help-36

[jagged flare]

[covert tree]

welp based on the edge case you provided

it seems like there should be a non-standard way to solving it

cuz it ain't symmetric

[jagged flare]

huh

wdym

[covert tree]

the proposed infimum is not when all numbers are equal

is what i meant by non-symmetric

u don't have a symmetric looking thing that achieve anywhere close to the minimum

[jagged flare]

yeah this problem is really funky

[covert tree]

idk i never had success using these big name theorem to get a min/max that is not symmetric/look nice

feeeeel like they work better in those cases

[jagged flare]

although theres a part 2 with \frac{x_n}{x_n+2x_{n+1}} which im not too excited about

[covert tree]

cuz many cases the min/max for those inequalities in those theorems is achieved when everything is equal

[jagged flare]

yeah thats what my intuition thought aswell

until it was betrayed

[covert tree]

welp try if the same thing works

intuition should tell us that the min should be 1 as well

[jagged flare]

yeah i will

hmm does it

[covert tree]

maybe not idk

cuz the first part of my solution would not carry through

[jagged flare]

ok i think it does

[covert tree]

if it is not increasing, the lower bound becomes 2/3

by my method

[jagged flare]

(the min being 1)

[covert tree]

not 1

[jagged flare]

wait wth

err

[covert tree]

well but there must be two increasing chunk max

cuz if there were more than 3, same argument applies and we have 3 chunks each greater than or equal to 1/3

[jagged flare]

playing around with the graph i couldnt get anything under 1

but it does apprach 1 more easily

[covert tree]

yeah it prob is 1

[jagged flare]

alright

[covert tree]

my argument prob works you just have to deal with 2 increasing chunk

some random ahh bound

[jagged flare]

possibly

im gonna try the problem now

i think ill close this, if i dont get it ill open another one cause i dont know how long ill take D:

thank you guyss

[covert tree]

welp gl

[jagged flare]

.solved ❤️

[final saddle]

Channel closed

Closed by @jagged flare

Use .reopen if this was a mistake.

[jagged flare]

tyy

im too used to instagram where double pressing means a like lmao

[final saddle]

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[dry kraken]

In metric space...d is distance function and it is always >=0

[dry kraken]

so why did they take range set as real numbers

[runic needle]

they not wrong

they could just add that d>=0 as an axiom

what does it say in quotes there?

[cosmic warren]

Yeah range \neq codomain

[final saddle]

@dry kraken Has your question been resolved?

[bold turtle]

@dry kraken Compare how you can write f: R - > R; f(x) = x²

[dry kraken]

here i will take all the values from R but range set would be >=0

Yeah we can take R

Thanks

.close

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[hard blaze]

Answer sheet says there are 3 images between D and G but I only count two, is there a mistake with the diagram or a mistake in my understanding?

[hard blaze]

<@&286206848099549185>

[iron frigate]

i think its saying D~E is 5ms, E~F is 5ms and F~G is 5ms so its three times

[hard blaze]

ohhh that makes way more sense now

thank you

do you have time for one small question related to kinematics?

I'lll post it anyway to see if another helper can help, for question a I used that s=ut+1/2*at^2 but I used the horizontal displacement instead of the vertical displacement for s and I got the wrong answer and I want to know why that is. The cannonball is travelling horizontally so why are we using the vertical displacement instead of the horizontal?

[iron frigate]

wait im not good at english ill help

[hard blaze]

nws take ur time and i appreciate ur help

[iron frigate]

do you know when you fire a cannon ball, the horizontal speed is same

[hard blaze]

same as?

[iron frigate]

bc it just gets gravity

[iron frigate]

sames as the first when you fire it

it doesnt change

[hard blaze]

ohh so the speed is the same but its going down because of gravity?

until it eventually hits the water

[iron frigate]

*the horizontal speed

the vertical speed changes

[hard blaze]

yeah right thanks for the correction.

[hard blaze]

and thats why we use the vertical displacement in the equation, oh okay now it makes sense.

[iron frigate]

so you just have to get the time while the cannonball falls 67m

[hard blaze]

And for question 3 why is the displacement not needed when calculating vertical velocity?

.close

[final saddle]

Channel closed

Closed by @hard blaze

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[strange pelican]

they had the time so they used that

[hard blaze]

so you can get the same answer using displacement?

[strange pelican]

you can

[final saddle]

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[mortal island]

Could someone check this for me?

[mortal island]

Here are the questions as well

There’s more to the assignment still but I want to make sure I at least have the first part right

[final tangle]

firstly you should be using () instead of []
and you've misidentified a few intervals

[mortal island]

Firstly…..where

[final tangle]

everywhere

[mortal island]

Im not sure that’s right….

[final tangle]

where you had a bracket, you should've had ()

[mortal island]

I don’t think that’s right-

Because it’s meant to represent included and excluded points

OMG TWINNNN

[blissful meadow]

I think their point is that generally "increasing" and "decreasing" are meant in the strict sense.

So not including stationary points

[mortal island]

Idk which one of us you’re talking about

[final tangle]

and you've misidentified stuff like from
-4 → 0
in Q2

[mortal island]

Which problem

[final tangle]

in Q2

[mortal island]

Quadrant two….what?????

[final tangle]

yes

in question two

[mortal island]

QUESTION two

Ok

[final tangle]

you've misidentifed the interval (-4,0)

[mortal island]

Hold on lemme look

[final tangle]

(2,4) in Question4

[mortal island]

OHHH I REMEMBER WHAT HAPPENED THERE

Ok ok I know why that’s wrong

Wait is the 4,inf right in q2?

The last interval?

[final tangle]

question 5 you have (1,0) which makes no sense,
having (0,1) doesn't align with the domain you had either

[mortal island]

Hold on hold on

Slow down

I’m still on 2

[final tangle]

that was wrong too

[mortal island]

Anything else in 2?

[final tangle]

no

[mortal island]

On

Ok*

The next issue is 4?

[final tangle]

what i've been able to spot, yes

[mortal island]

Ok what’s the issue with 4

[final tangle]

double check whether your intervals are increasing/decreasing

[mortal island]

Ok gimme 2 seconds

Is it the 2,4?

[final tangle]

yes

[mortal island]

Ok ok

Any other issues there?

[final tangle]

q4, no

multiple issues after that

did q5→8 ask for extra stuff?

[mortal island]

Figures cus that’s where I had no help and the types of questions changed

[mortal island]

Yes it did

In the picture for 5-8 it has the new directions as well

[final tangle]

question 5 you have (1,0) (as an interval) which makes no sense,
having (0,1) doesn't align with the domain you had either

for intervals of increase / decrease you should've had
(-inf, 0)
(0,inf)

[mortal island]

Interval…?

Wait hold on

Lemme read that again

[final tangle]

what you've been doing in questions 1→4

[mortal island]

Ok

Can you explain…why that is

[final tangle]

you've yourself identified the domain to be (-inf, inf)

[mortal island]

Yes

[final tangle]

so why are you starting your interval at
what i'm assuming was intended to be -1
instead of -inf

and stopping at 1 in (0,1) instead of going to inf

[mortal island]

So it’s the interval that’s wrong and not the domain/range right?

[final tangle]

yes, that's what i said

[mortal island]

I just wanna make sure

[final tangle]

the interval for increase/decrease

[mortal island]

Any other issues with 5?

[final tangle]

no

[mortal island]

Right anything else?

[final tangle]

Q6 has many errors,
that i'll get you to redo that question again

[mortal island]

I’m just gonna have you look over 7 and 8 too so I can erase once and not 3 times

[final tangle]

q7, (0,-1) and (0,1) aren't intercepts, the curve is just very close to the y-axis

it wouldn't be a function, and you could do a quick check by plugging x=0 into the function

[mortal island]

So just erase the whole thing?

I know 8 isn’t right there’s no way

[final tangle]

Q7 just erase the extra intercepts

other parts of that question are fine

Q8, domain and range are wrong
and intervals of increase/decrease are wrong, same issue as before, why stop and start at -1 and 1

[mortal island]

Domain and range are wrong?

[final tangle]

yes, that's what i said

[mortal island]

How do you see that-

[final tangle]

what's your reasoning for putting (-inf, inf) for both?

[mortal island]

It extends indefinitely upward and extends indefinitely to either side-

[final tangle]

extends infinitely upward, yes,
but is the graph extending infinitely downward anywhere?

[mortal island]

If it gets long enough I would think so

It looks like it should

[final tangle]

it will not
didn't spot it earlier, but the end behaviour was also wrong

the graph here has what are referred to as asymptotes

[mortal island]

So is it….. 0,inf?

I know nothing about asymptotes

[final tangle]

yes, but becareful whether you use a ( or [ for 0

[mortal island]

Uhhhh it does not include zero….i think

[final tangle]

yeh, it will not include 0 here

[mortal island]

Is the domain alright?

[final tangle]

algebraically,
$$\frac{8}{x^{4/5}}$$
as $x$ gets large, that'll get closer and closer to 0

[soft zealot]

ραμOmeganato5

[final tangle]

no, as mentioned earlier, domain was also wrong

is the function defined at x=0?

it should be clearer in the above fractional form

[mortal island]

Uh….no

[final tangle]

thus that shouldn't be in your domain

[mortal island]

Right so…. Is the domain like…. Split?

[final tangle]

yes

this was the same issue as Q6

[mortal island]

Are the intercepts ok?

[final tangle]

for Q8 yes

[mortal island]

Ok let me fix the top line on 8

So (-inf, 0) (0,inf)?

[final tangle]

yeh

you'd want a $\cup$ in between

[soft zealot]

ραμOmeganato5

[mortal island]

I don’t think we’ve used that before….

[final tangle]

essentially states the combination of your two intervals

[mortal island]

Ok and what’s up with the end behavior

[final tangle]

as mentioned earlier, the graph doesn't shoot down to -inf

but instead gets closer to 0

[mortal island]

So that one is….. (0?

[final tangle]

no,

[mortal island]

I’m a little lost

[final tangle]

as x→-inf, f(x)→0

is the end behavious component

x→inf, f(x)→ 0

[mortal island]

Just….0?

[final tangle]

yes

[mortal island]

Wait hold on

And the regions of increase and decrease?

[final tangle]

similar issue as before
why stop at -1 and start at 1

[mortal island]

Is it 0?

Cus it goes past -1 and 1 but not…in the space provided on the graph

[final tangle]

the arrow indicates the graph keeps going

[mortal island]

Yeah

[final tangle]

it'll will keep going until 0, (but won't reach as that isn't in the domain)

[mortal island]

so exclude zero?

[urban raptor]

Yep.

[mortal island]

Oh hello

Right so back to our…….problem area…. 😬 6

[final tangle]

with all the problems i've pointed out for 5,7,8
you should now be able to do Q6

(if you've understood everything and fixed those mistakes)

[mortal island]

Im gonna talk though it

So the domain I’m thinking is -inf to 0 and then 0 to inf? Because I assume it’ll continue to go inward until it’s ALMOSF zero but not

Is that what an asymptote is?

Range would be….0,inf since it doesn’t go below zero… at least that I can tell

Intercepts…I’m not 100% sure on this one can’t lie

It doesn’t intercept y I know that

But I can’t….really tell what the x intercept should be

[final tangle]

you need to show whether you're using () []

[urban raptor]

x-intercept is when y = 0.

[mortal island]

Does that matter in domain and range?

[mortal island]

I know that-

[final tangle]

whether something is included/excluded matters everywhere

[mortal island]

Oh shit

So (-inf, 0) u (0,inf) for the domain

And [0,inf) for range?

[urban raptor]

Almost, the graph never touches x=0.

[mortal island]

So it’s not included

So (0,inf)

[urban raptor]

Yes.

[mortal island]

Ok so what about these intercepts?

Cus it looks like it’s touching zero…everywhere

[final tangle]

again recall what you just answered for range and why

[mortal island]

But it never touched zero…so…..no intercepts…?

[final tangle]

yes

[mortal island]

Yessirrrr

End behavior ok ok

I’m thinking x -> -inf , f(x) -> 0 and the same thing for positive?

It’s not approaching infinity because it’s not going up

It’s not approaching negative infinity because that isn’t in the range

So 0?

[final tangle]

yes

you're also giventhe function equation
so you don't need to soloely rely on the graph

[mortal island]

Do you not need to handle this..

Peak

Lmao

I’m a more visual person that equation- is freaking me out

[compact laurel]

Visuals are the GOAT’s

[mortal island]

I’m thinking….(-inf,0) it’s increasing and (0,inf) its decreasing?

[final tangle]

yes

[mortal island]

Alright!

So now we have 9 and 10 which are a different type of question

And I’m…not sure how to do these

[compact laurel]

It’s asking you to describe in words, yeah?

[mortal island]

I presume

[compact laurel]

Well, let’s start with orientation. How can we at least get the graphs to point in the same directions

(For 9)

[mortal island]

Change the direction

[compact laurel]

Well, yeah…. lol. But how

[mortal island]

I don’t know

[compact laurel]

Well, usually, it’s as easy as reflecting the graph about an axis

[mortal island]

That’s how it’s phrased

Ok

[compact laurel]

Can we do that for 9? Reflect about an axis (x or y) and get them pointing in the same direction?

[mortal island]

Y

[compact laurel]

Reflect about the y?

[mortal island]

Yes

[compact laurel]

Nice!!

So, that gets them point in the same direction

But now to (stretch or compress) the graph

Horizontal stretch and compression, or vertical stretch and compression

Do you know what those are?

[mortal island]

Horizontal stretch

By a factor of……3?

[compact laurel]

Do you have to state the exact amount?

[mortal island]

I presume so….

Looks like 3 to me

[compact laurel]

Why does it look like 3?

[mortal island]

Where it gets to one

It changes from 1 unit to 3 units

Where y=1 on the normal one it’s x=1 but on the dotted one it’s x=-3

[compact laurel]

Bingo!!!

[mortal island]

Ok so for 10…

[compact laurel]

And does it want you to state the function afterwards, or no?

[mortal island]

I have no clue

So I’m not Finn

Gonna**

[compact laurel]

Alr!!

So let’s start with orientation

[compact laurel]

How do you make them face the same direction for 10?

[mortal island]

Rotate it…

90° clockwise…

[compact laurel]

Remember, it’s usually as easy as a reflection

[mortal island]

Then reflect it over the x axis I guess

[compact laurel]

Yes! Funny thing here, in this case a reflection about the x is the same as reflecting about the y. So it doesn’t matter here. lol

So, once we’ve reflected it, are the any stretches or compressions?

[mortal island]

I don’t think so

And it doesn’t need to be translated

[compact laurel]

Neither do I, lol

[compact laurel]

Are we sure?

[mortal island]

No I was looking at it wrong

It needs translated up….3

I don’t think it needs to go left or right

[compact laurel]

Yessir!!!

[compact laurel]

And I would agree

[mortal island]

Right

I’ll be back but I gotta go get dressed

.close

[final saddle]

Channel closed

Closed by @mortal island

Use .reopen if this was a mistake.

[final saddle]

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[mortal island]

I’m back! Could anyone help with some of these?

[rich swallow]

have you tried to graph them out?

[mortal island]

I can’t for this assignment

I know for 11 it reflects over…I presume the x axis

And it’s stretched….in some direction by 3

How do I determine wether it’s horizontal or vertical-

Because for this problem it does matter…

[acoustic oar]

the transformation multiplies the output of the function by 3

so the values become more extreme

[mortal island]

Right I get that

[acoustic oar]

so its a vertical stretch

[mortal island]

Ok-

I have 12 done

So for 13- I have to apply the changes to the function

So I think…..it would be 3(1/x -2)?

The 3 makes it stretch and the -2 makes it go right….

[acoustic oar]

I think its right yea

<@&268886789983436800>

[mortal island]

????

What happened

[acoustic oar]

there was a bot with scam

[mortal island]

Ah

14…I’m not too sure what to do

[acoustic oar]

what is the reflection about y-axis?

[mortal island]

Uhm…like- the negative version of the original function..?

[acoustic oar]

that would be -f(x)

[mortal island]

Ohhhh

So…my answer would be -f(x) = - sqrt x + 2?

(2 not included in the square root)

[acoustic oar]

this is f(-x) not -f(x) which is the reflection about x - axis

[mortal island]

Wait huh

[acoustic oar]

this is not right

you should try to draw

[mortal island]

I can’t

I don’t have anything to draw it on

Nor do I know what this function looks like

[acoustic oar]

open desmos.com

[mortal island]

I’m not allowed to use that

[acoustic oar]

why not

[mortal island]

Or I’d have this assignment done by now

[mortal island]

My teacher is a psycho

She doesn’t believe in subtraction

Or division

[acoustic oar]

then take some exercise you have already done and play with it in desmos

[mortal island]

Or making life easier

[acoustic oar]

if you know how these transformations look like you will have an easier time thinking about them and applying them

[mortal island]

Or you could just…explain why it doesn’t do what it does

[acoustic oar]

okay

if you reflect about y - axis that means that every point is symmetrical wrt this y-axis so f(x) = f(-x) and the reflection about the x - axis takes the function output and flips the sign so its -f(x)

its not cheating if you just go to desmos and apply transformations that way you can even see if what you are doing is correct

[mortal island]

So it’s….-f(-x)?

[acoustic oar]

yes thats the first transformation of 14

[mortal island]

And it equals (sqrt x) + 2

[acoustic oar]

the result?

no not quite

[mortal island]

Wait no

It’s sqrt x +2

Just without the parentheses

But the 2 isn’t under the square root

[acoustic oar]

where did the reflections go about x and y axis

[mortal island]

On the first part???,

Like you said???,

[acoustic oar]

we do f(-x) then 2f(x) and then -f(x)

[mortal island]

????

[acoustic oar]

wdym

[mortal island]

I’m not picking up what you’re putting down

[acoustic oar]

we have $\sqrt{x}$ first transformation takes does f(x) -> f(-x), so we get $\sqrt{-x}$

[mortal island]

Ok..

[soft zealot]

rose

[acoustic oar]

,w graph sqrt(x)

[soft zealot]

Results provided by WolframAlpha

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[acoustic oar]

,w graph sqrt(-x)

[soft zealot]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by lavenderroseee_

[mortal island]

Wait is it…

-sqrt -x +2?

[acoustic oar]

lets see

,w graph -sqrt(-x) + 2

[soft zealot]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by lavenderroseee_

[acoustic oar]

not quite its not stretched

+2 transforms the output up by 2 units

oh wait

nono mb i misread

nvm

[mortal island]

So is that right?

[acoustic oar]

its like I said

no

[mortal island]

It’s not -sqrt -x + 2?

[acoustic oar]

you added +2 to sqrt(-x)

lets see what this does

,w graph sqrt(-x)

[soft zealot]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by lavenderroseee_

[acoustic oar]

,w graph sqrt(-x) + 2

[mortal island]

Hold on lemme reformat it

[soft zealot]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by lavenderroseee_

[mortal island]

(-sqrt -x) + 2

[acoustic oar]

we moved it 2 units up

we wanted to stretch so not quite what we wanted

[mortal island]

So…..

2(-sqrt -x)

[acoustic oar]

yes thats right

[mortal island]

Ok

15 and 16….i don’t know how to do

[acoustic oar]

set y = f(x) for brevity

and manipulate this

so you get x = (something in terms of y and not in terms of x)

but idk maybe they want you to use transformations

[mortal island]

Huh…

It wants the inverse

[acoustic oar]

yes, but we can do this in two ways algebraically or using transformations

[mortal island]

I don’t know…

[acoustic oar]

so algebraically it would be like this, suppose we had to find inverse of $y = x^3 + 5$ then I want to manipulate to have only x on one side, so we would do $y - 5 = x^3$ and then $\cbrt(y-5) = x$, so our inverse is $f^{-1}(x) = \cbrt(x-5)$

[soft zealot]

rose
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[blazing summit]

think about solving for x

[acoustic oar]

yes this is the idea

[mortal island]

So am I just…solving for x

Or y rather

[blazing summit]

essentially, first solve for x, in terms of y, then you just swap the variables after

[mortal island]

….what

[blazing summit]

ill help you get started. On 15, add 2 to both sides giving you f(x)+2 = -2/(x+2)

you can rewrite f(x)= y

just for clarity sake

[mortal island]

What does “solve for x in terms of y” even mean…..

[blazing summit]

basically, x= some equation with y

[mortal island]

Ok

[blazing summit]

so, if we wanna solve for x, can you think about what to do next after we added 2 to both sides?

[mortal island]

Multiply by the denominator

[blazing summit]

correct

[mortal island]

Right so now I have y+2 (x+2) = -2

[blazing summit]

good

[acoustic oar]

parentheses around (y+2)!

[mortal island]

I know

[blazing summit]

so now what do we divide both sides to further isolate x?

[mortal island]

Divide by…y+2?

[blazing summit]

yup

[mortal island]

And then subtract 2

[blazing summit]

yes

[mortal island]

So x= -2/y+2 -2

So I did all the math for no reason

Because it literally just…swaps

Is that in all cases or just this one?

[blazing summit]

yeah, for 15 it’s one of those weird self‑inverse ones so solving it just lands you back on the same formula. the ‘swap x and y’ thing is literally what the algebra is formalizing, so at least now you’ve seen why it swaps in this case lol

so the inverse is -2/x+2 -2

you just swap x and y at the end

[mortal island]

Interesting

So 16

Start by adding 3 on both sides

[blazing summit]

yes

[mortal island]

So y+3 = (x-2)^5

[blazing summit]

good

[mortal island]

How do I get out of that situation

[blazing summit]

you take the 5th root of both sides

[mortal island]

So 5throot of y+3 +2 = x?

[blazing summit]

yeah

[mortal island]

Ok!

[acoustic oar]

whats the inverse of 1/x?

[mortal island]

That’s the last of this assignment but I still have more to do

[blazing summit]

dont forget to swap y and x at the end

[mortal island]

Oh gotcha

[blazing summit]

so the inverse would be 5th root of (x+3) + 2 = f^-1(x)

[mortal island]

Perfect gotcha

Alright now I’ve got to work on- asymptotes

I have actually no clue what those are or how to do them

[blazing summit]

so an asymptote is basically a line the graph gets closer and closer to but never reaches

[mortal island]

Ok…

[blazing summit]

i can provide an image of one to help you visualize it better

[mortal island]

No I know what the definition is

I don’t know what I’m doing

[blazing summit]

okay

[mortal island]

I couldn’t look at that equation and tell you what anything is or how to find anything

I missed the whole unit on asymptotes

[blazing summit]

You factor the top and bottom, cancel any common factor (that gives holes), whatever’s left in the denominator gives vertical asymptotes, and then you look at the degrees to get the horizontal/slant one

[mortal island]

Uh….

You’re gonna have to help me out w that one

[blazing summit]

alright

lets look at number 1

[mortal island]

Ok

[blazing summit]

so i want to make this easier to look at so i can infer more about function

one way to do that is to factor the denominator

are you comforable with that?

[mortal island]

You…can’t factor the denominator-

It’s in the most factored form

[blazing summit]

you actually can factor it

[mortal island]

Oh wait

Ok Ik what you’re saying

One sec

-x+1/ (x-3)(x-1)

[blazing summit]

yes correct

[mortal island]

But -x+1 is Essentially……x-1?

[blazing summit]

no, -x+1 = 1-x

but anyways

you have those two terms in the denominator

[mortal island]

Oh yeah

Can it be like…factored out…?

[blazing summit]

to find the vertical asymptote, look at the denominator, and find the values of x that makes the denominator 0

[mortal island]

So….3 and 1?

[blazing summit]

yes

[mortal island]

So is it just….

Like- how do I write that

[blazing summit]

so what this tells you is the vertical asymptotes are at x= 3, and x=1

for the horizontal one, the top has degree 1 and the bottom degree 2, so as x→±∞ the fraction goes to 0 → horizontal asymptote y=0.

[mortal island]

…what

[ripe jewel]

do you know the definition of a horizontal asymptote

[mortal island]

I know nothing about asymptotes

I’m assuming the horizontal one is on the x axis

[ripe jewel]

well learning about them would be a good place to start

[mortal island]

Well I don’t have time for that

I have to turn this in so I don’t fail my class

[ripe jewel]

ah,that's rough. good luck with that

[mortal island]

Yeah…

The school system is designed to make children fail if they’re ill

[blazing summit]

if i were you, i would play with desmos a little and test different functions

[mortal island]

I’m not allowed to

My teacher is psychotic

[ripe jewel]

I don't think you'll get very far by being bitter, though. maybe you should try learningt about asymptotes, they're really not very complex

[mortal island]

I’m not bitter she’s actually crazy 😭😭😭 she doesn’t believe in subtraction or division

[ripe jewel]

nevertheless

[mortal island]

She has all these weird rules about formatting that no one understands which makes it like impossible to get help

[ripe jewel]

this server is to help you learn stuff, not to give you answers without explanation, so, it is what it is

want me to teach you?

[mortal island]

I’m not looking for answers with no explanation I’m looking for guidance

I would like that yes

[ripe jewel]

sure okay

[mortal island]

I just have no prior knowledge

[ripe jewel]

let's start with horizontal asymptotes because they're easier

[mortal island]

Sure

[ripe jewel]

familiar with limits?

at all?

[mortal island]

Uhhhh i think it’s like- a graph can get extremely close to a certain point but not pass it?

[ripe jewel]

that's not a bad start

here we're dealing with limits at infinity

it tells you the "end behavior" of a function as the input gets very very large

or very very small (negative)

$$f(x) = \frac{x}{x+1}$$

[soft zealot]

gfauxpas

[ripe jewel]

example:

if you put in x = -100, x=-1000, x = -10000000

do you see how this function becomes very close to a horizontal line?

[mortal island]

Yeah

[ripe jewel]

so a horizontal asymptote is a horizontal line that a function "acts like" when x approaches +infinity or -infinity

a function need not have a horizontal asymptote at all, or it can have 2

[mortal island]

so the options are 0 or 2?

It can’t have 1?

[ripe jewel]

or 1

if the limit at positive infinity and the limit at negative infinity act the same

[mortal island]

Ok-

[ripe jewel]

so, try it for your exercise

$$f(x) = \frac{-x+1}{x^2-4x+3}$$

[soft zealot]

gfauxpas

[ripe jewel]

the bottom grows much faster than the top

[mortal island]

Yeah-

[ripe jewel]

so, whether towards +infinity or negative infinity

it just acts like what horizontal line?

[mortal island]

I’m lost-

What am I trying to do

[ripe jewel]

trying to see if the function acts like a horizontal line for very large x or very negative x

[mortal island]

Right and how do I do that-

[ripe jewel]

well for any function where the denominator grows much faster than the numerator

it's just going to go to zero pretty fast right

[mortal island]

I guess-

[ripe jewel]

well, try it

f(10000000)

numerator: 10000000-1

denominator: 10000000^2 - 4(10000000)+3

the denominator is huge

much larger than the numerator

[mortal island]

Ok…

I see that

[ripe jewel]

so as x->+infinity or x->-infinity, this function is going to be approximately zero

in other words, approximately the horizontal line y = 0

[mortal island]

Ok…

I’m not sure that I’m following

[ripe jewel]

we say that y=0 is a horizontal asymptote of f, because f(x) has a constant limit (0) as x->infinity or x->-infty

all that means is

as x is very large, f(x) is essentially constant, 0

[mortal island]

Ok-

[ripe jewel]

there's another problem on your page where f has a horizontal asymptote y = 0, which is it?

[mortal island]

Is that not what I said earlier? That a horizontal asymptote was always gonna be at y=0?

[ripe jewel]

which function acts like y=0 as x->infinity

[ripe jewel]

not always

consider $\frac{x}{x+1}$

[mortal island]

4..?

[soft zealot]

gfauxpas

[ripe jewel]

has a horizontal asymptote at y=1

[mortal island]

I’m really not sure that I’m following

[ripe jewel]

at which part did you get lost

[mortal island]

All of it

I’m a very visual person-

[ripe jewel]

oh, sure, let's do a graph

[mortal island]

Ok-

[ripe jewel]

as x gets larger and larger, the function becomes almost a horizontal line at 0

[mortal island]

Yeah

Is that not…also what I said earlier 😭😭😭😭 it can get super close but never touch?

[ripe jewel]

yeah but it's not always at y=0

here's the x/(x+1) example

it approaches the line y=1

[mortal island]

Ok-

[ripe jewel]

now, here's the trick

when you have a fraction where the numerator and denominator are both polynomials - know what those are?

[mortal island]

Difficult-

[ripe jewel]

a polynomial is an expression like 5x^4+3x^2-x+1

all powers of x are whole numbers and multiplied by numbers

[mortal island]

Yeah I know that

[ripe jewel]

okay

so when you have a fraction of polynomials

[mortal island]

It’s difficult hahah I’m so funn

[ripe jewel]

the horizontal asymptotic behavior only depends on the terms with the highest power

so for example

$$\frac{-x+1}{x^2-4x+3} \sim \frac{-x}{x^2}$$

[soft zealot]

gfauxpas

[ripe jewel]

the reason is that when x becomes extremely large, those terms are so much bigger than the other terms that they determine the behavior of the whole function

[mortal island]

Ok…

[ripe jewel]

-x/x^2 = -1/x, and is it obvious to you that this acts like 0 as x->infinity?

[mortal island]

I guess…?

[ripe jewel]

well, you can look at the graph, or you can try putting in numbers

if you're not convinced

[mortal island]

Is there like a crash course type video you could show me so I don’t have to waste your time?

[ripe jewel]

youre not wasting my time, but you can try khan academy

[mortal island]

Ok brb

What would be a good topic name to search. Up

[ripe jewel]

infinite limits

asymptotes

[mortal island]

Ok

[ripe jewel]

https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:rational-functions/x9e81a4f98389efdf:graphs-of-rational-functions/v/graphs-of-rational-functions-horizontal-asymptote

[mortal island]

Oh cool

[ripe jewel]

actually i would start from here https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:rational-functions/x9e81a4f98389efdf:end-behavior-of-rational-functions/v/end-behavior-of-rational-functions

gl

[mortal island]

I don’t know that I love his explanations especially since he did the entire problem wrong- I’m gonna try a different explanation and brb

[ripe jewel]

it doesnt change the answer, focus on the method

or explore other videos

idk

[mortal island]

IMO it most likely did change the answer

But I didn’t work out the original problem

[ripe jewel]

it does not change the answer

in this case

[mortal island]

We’re gonna try organic chemistry tutor’s all encompassing video on….basically everything the assignment is asking

[final saddle]

@mortal island Has your question been resolved?

[mortal island]

Ok so I think I get the first part- but I still need to figure out the other parts

[final saddle]

@mortal island Has your question been resolved?

[mortal island]

Ok now that I have a rough concept of what’s happening…. <@&286206848099549185>

@ripe jewel I’m backkk

Helloooo?

[kindred mortar]

can you recall what you were trying to do ?

[mortal island]

Oh yes

[mortal island]

^^

[kindred mortar]

1. ?

[mortal island]

Yes

Let me show what I have so far-

[kindred mortar]

ok you factorized the denominator. Now you have do determine if the function as a vertical asymptote in 1 and 3 or not

Do you know how to do it ?

[mortal island]

The vertical asymptotes are on the side of the page

That’s the VA

[kindred mortar]

oh sorry didnt read well

do you know how to rotate documents ?

[mortal island]

Uhhhh

,rotate

[mortal island]

HAH

[kindred mortar]

perfect

[mortal island]

And HA is the horizontal asymptote

[kindred mortar]

ok so it is not correct for va

[mortal island]

Oh- how is that?

It cancels out the bottom to 0

[kindred mortar]

yes but numerator goes to 0 aswell in the case of 1

[mortal island]

How do you figure that,

[kindred mortar]

replace x by 1 in the numerator

[mortal island]

Ohhh no I see

So would it just be…3

[kindred mortar]

ok so this is not that simple for this cas

[kindred mortar]

3 is correct

because when x goes to 3 denominator goes to 0

but numerator stays bounded

[mortal island]

Wait…no it doesn’t-

[kindred mortar]

so yes 3 is a va

[mortal island]

Cus it’s (3-3) and (3-1)

Which would leave 2 in the denominator

Would it not?

[kindred mortar]

multiply them

0 * 2 = ...

[mortal island]

Ah I see

[kindred mortar]

so 3 is va

no problem

[mortal island]

Ok

[kindred mortar]

but 1 we dont know yet

and now i dont really now what tools you are supposed to use

[mortal island]

Is 1 a hole then? Since I cancels both the top and bottom?

[kindred mortar]

do you know limit ?

[mortal island]

Vaguely yes

[kindred mortar]

it can be hole or a va but w don't know yet

Ok the technic i would use here

[mortal island]

Well it’s not a VA because it cancels the top…

[kindred mortar]

factorize num and denominator by (x-1)

and cancel them

for $x \neq 1$

[soft zealot]

Lin Xia

[mortal island]

Uhh…

[kindred mortar]

it is not that simple

[mortal island]

It should be 🥲

This is the first problem

Factories it where x ≠ 1?

[kindred mortar]

you can have numerator goes to 0, denominator goes to 0 but fraction goes to 2 for example or +inf

[kindred mortar]

the problem is simple but we have to be a bit cautious

if you factorize what do you get ?

[mortal island]

Uhhhh…let me figure that out

…?

I really have no clue on tbat

[kindred mortar]

ok dont forget brackets

[mortal island]

What

[kindred mortar]

around x-1 like that (x-1)

parentheisis ?

[mortal island]

But you only use that for the inside….

[kindred mortar]

i dont know the word

[mortal island]

You only use that for the factored part- what’s left-

[kindred mortar]

and your denominator should stay the same right ? It is already factorized

[mortal island]

I guess

[kindred mortar]

not convinced ?

[mortal island]

No not that just confused

It makes sense in theory but I don’t get why….

[kindred mortar]

you dont get why what ?

why we do that ?

[mortal island]

Cus don’t you have to do it on both the top and the bottom?

Like whatever you do on the top you also have to do on the bottom

[kindred mortar]

you confuse with something else

[mortal island]

Ok I guess

[kindred mortar]

you can multiply a fraction on num and denom by the same thing

but that's not what we are doing here

[mortal island]

Ok

Do the top and bottom cancel?

[kindred mortar]

we are factorizing numerator so that a factor (x-1) appears and cancels with the denominator's one

show me what you got now

[kindred mortar]

almost perfect

you should have brackets on the num aswell

[mortal island]

that’s not how my teacher does it but I’ll trust you

[kindred mortar]

because if you expend now x - 1(-1) you'll have x + 2 that is not equal to -x + 1 right ?

[mortal island]

No…

That’s not how that works

[kindred mortar]

oh ?

[mortal island]

1x1 doesn’t equal 2

[kindred mortar]

lol

y mb

+1 sorry

[mortal island]

Lmao

Either way lol

[kindred mortar]

x + 1 still not equal to -x + 1

[mortal island]

Fair enough

I fixed it w the parentheses

[kindred mortar]

perfect

and now you have the same factor on num and denom

[mortal island]

Do the x+1’s cancel?

[kindred mortar]

so you can cancel them

[mortal island]

That’s some sort of hole yes?

[kindred mortar]

when x\neq 1

and yes

[mortal island]

What-

[kindred mortar]

x not equal to 1

[mortal island]

In what context

[kindred mortar]

what we are doing

we are supposing that x is not equal to 1

[mortal island]

What is that for

[kindred mortar]

because in 1 our expression is not defined

[mortal island]

And how do I write…the hole…?

Just x-1?

[kindred mortar]

since you devide by (x-1)

[mortal island]

Yeah I got that

[kindred mortar]

okok

[mortal island]

But what part of the question does it apply to

[kindred mortar]

so now you have a new expression

1/(x-3)

[mortal island]

Yes

[kindred mortar]

and when x goes to 1

this stay bounded

so you can say that 1 is a hole

and not an asymptote

[mortal island]

Just….1?

[kindred mortar]

yes

[mortal island]

Isn’t a hole supposed to be a coordinate?

[kindred mortar]

hum

can be

[mortal island]

It has to be if it’s on a coordinate plane

[kindred mortar]

ok so 1 is the first coordinate

what is the second one you think ?

[mortal island]

Uhhhh

You’d have to plug something in-

[kindred mortar]

it is the value that f is supposed to take on 1

[mortal island]

Right so y

But we don’t know y

[kindred mortar]

use what we did so far

you have a simplified expression of f

what does it give when x goes to 1 ?

[mortal island]

So…plug in 1 for x?

[kindred mortar]

yes but not in the original version of f

because we saw that it doesn't work

[mortal island]

Yeah that’s what I mean

[kindred mortar]

so use the simplified version we got

[mortal island]

-1/4?

[kindred mortar]

not exactly

show me your calc

[kindred mortar]

ok almost

you did put x = -1

but it is x = 1

[mortal island]

OH

So -1/2?

[kindred mortar]

not exactly either

but we are close

[mortal island]

1-3=-2

[kindred mortar]

yes

[mortal island]

And -1 on top…..

[kindred mortar]

yes

[mortal island]

So……

What’s wrong

[kindred mortar]

-1/(-2)

[mortal island]

That means the same thing

[kindred mortar]

is the same as 1/2

not -1/2

[mortal island]

So….

(1,1/2)?

Right

So we have a hole, a vertical asymptote and a horizontal asymptote…

We still need x and y intercepts and intercepts at vertical and “slant” asymptotes

[kindred mortar]

do you know how to do ?

[mortal island]

No…

[kindred mortar]

i'm not really sure to understand what it means so tell me if i'm wrong. x and y intercepts ar for asymptotes ?

[mortal island]

I’m not 100% sure…

[kindred mortar]

let's assume it is

you have one vertical asymptote, what is its y intercept ?

x intercept sorry

[mortal island]

I might redo this tomorrow and ask my teacher since I’m not…rly sure what most of this is asking and/or what I’m supposed to be doing

[kindred mortar]

ok we can do asymptotes for questions 2,3,4 if you want

[mortal island]

I might just switch assignments to optimize my time

[kindred mortar]

okok

[mortal island]

New issue- proofs

And it’s been a hot minute since I’ve done proofs

<@&286206848099549185>

[timber basin]

Hey, I would try to go from one side to the other

to prove that they are equal

[mortal island]

You can’t cross sides

That’s like the number 1 rule of proofs

[timber basin]

look at 1 for example: 1/(cot^2 x) is equal to tan^2 x

[mortal island]

No it’s not

Wait yes it

Is

But I don’t have 1/cot^2

I have cot^2 by itself

[timber basin]

its the denominator

so you can pull it out of the expression

2sinxcosx / cot^2 x = (2sinxcosx)(1/cot^2 x)

[mortal island]

Idk about that-

[timber basin]

its multiplicative so it works,

[mortal island]

But I could also just….change it to 1/tan^2

Or I could change it to csc^2 -1

There’s a lot that could be done here but I’m not sure what benefits me the most

[mortal island]

I can’t see fha

[timber basin]

sorry

this is how you pull out 1/ cot^2x

and then we know by definition that cot x = 1/ tanx right?

so 1/cot^2x = tan^2x

[mortal island]

I just said that

[timber basin]

yeah

[mortal island]

And gave you another way to change it as well

Right so now that I’m here….

[timber basin]

then use the double angle identity

2cosxsinx= sin2x

[mortal island]

Forgive me it’s been a few months since I’ve worked with proofs

[timber basin]

haha, no worries bro

[mortal island]

There’s still more

Just give me a sec to write it down

[timber basin]

for proofs in general, i'd just try to manipulate the left side to match the right one. It should work generally quite well in these eqs 🙂

[mortal island]

Ok so for number 2

The only thing I’m seeing applicable would be power reducing formulas

Only problem is….nothing is over 2

[timber basin]

this one, i would look for the other double angle identity

you have cos2x-->sin^2 (x) in some way

and cos^2 (x) too

[mortal island]

What

[timber basin]

well, you have to reduce the argument of the cos function

and you generally do that through a double angle identity

[mortal island]

What????

What are you on about…

[timber basin]

these identities come in very handy when doing these proofs

[mortal island]

I have all of these

The papers are sitting beside me

What the fuck does “the argument of cos” mean

[timber basin]

cos(argument)

basically just whatever is in the brackets

[mortal island]

Again…huh

Where are you seeing brackets anywhere in this equation…

[timber basin]

well, for sin and cos and those kinds of functions, the brackets are implied

sorry for the confusion

[mortal island]

I guess

[timber basin]

anyway, let's use the cos(2theta) identities here

[mortal island]

Could you not…manipulate the RHS using the powder reducing formulas..?

[timber basin]

you can do that too! It just depends on your preference

I generally go from left to right, but you can go from right to left as well

just don't mix and match

[mortal island]

I know

That was just the first thing I saw that looked similar

[timber basin]

I feel like maybe cos(2theta) = 1- 2sin^2(theta) could be useful here. Try putting it into your numerator

[mortal island]

Ok but…what about the 1 out in front

[timber basin]

it cancels out!

[mortal island]

No….?

Cus you have to prove that it cancels out

Everything has to be explained

[timber basin]

and for the bottom, i'd use cos(2theta) = 2cos^2(theta)-1

[mortal island]

It can’t just happen for no reason

But you can’t…do that

You can’t just say it cancels for no reason

[timber basin]

here

[mortal island]

What….

[timber basin]

step 2, we substitute in the double angle identities

for cosine

[mortal island]

I don’t think you can do that…it doesn’t coincide with any of the identities

[timber basin]

cos (2theta) = (1-2sin^2x) is given as an identity in the sheet

[mortal island]

Yeah but you can’t do that if the 1’s are attached…

[timber basin]

ahh, i see what you mean, the 1s are basically not a part of it

you can substitute it as a part of the equation actually

[mortal island]

Then why did I get points taken off the last time I did it like that 🥲

[timber basin]

I don't know, perhaps it was a similar siutation with a slightly different syntax

[mortal island]

Ugh why is this shit so confusing

[timber basin]

hey, i get it man, it's confusing, trig's my least favourite part of math actually

[mortal island]

There are no fucking numbers

Math is numbers

[timber basin]

haha

[mortal island]

This is a vague concept

It’s like the fucking Lacroix of math

Like you ate a strawberry and farted in the can- that doesn’t mean it’s strawberry flavored

[timber basin]

I'm not gonna interrupt your tangent here, but yeah, it's kinda hard

but it gets much easier when you just accept that okay, i guess this is how it is

[mortal island]

Point being math is hard and Lacroix is nasty

[timber basin]

and just use the identities like religiously

you wanna go for nr. 3?

[mortal island]

No

I’m not even grasping the concept of 2

[timber basin]

cool, okay

[mortal island]

Also I think your identities are backwards

[timber basin]

let's work through it?

[timber basin]

haha

[mortal island]

Yes please

[timber basin]

okay, well, using my backwards identities, since they are the only one i have

[mossy inlet]

wait i just got here, whats the problem ? (sorry for inconvenience; ill try to help)

[tepid valley]

Fr it's so far up.

[timber basin]

proofs

[mortal island]

You better not be a freshman that knows more than me I will start crying

Genuine tears

[tepid valley]

Oh which one?

[timber basin]

i'm omw to college bud dw

[mossy inlet]

ah which number problem?

[timber basin]

nr. 2

[mossy inlet]

nono my class is also on this

[mortal island]

Not you