help-13

[snow grail]

Let’s start with the angle between the positive y axis and the negative x axis, how many degrees are in that angle?

[upbeat pond]

Idk 90

[snow grail]

Yep, the x and y axis make a right angle

[upbeat pond]

Oh okay i was overthinking that question

[snow grail]

No worries

Now, how many degrees does the line from red to blue take away from this angle?

[upbeat pond]

20

[snow grail]

And how many are left?

[upbeat pond]

70

[snow grail]

Now use the parallel angle theorem with the horizontal lines to find the interior angle at blue

[upbeat pond]

Okay

Wait what

Is it 110 ? sorryim confusing

[snow grail]

Let me draw out what we just did to just red

Here’s where we are now, put that at red on your diagram

And use the parallel angle theorem using horizontal lines this time

[upbeat pond]

So we r trying to find interior of blue angle

[snow grail]

Yes

[upbeat pond]

?

[snow grail]

Exactly

That’s correct

Now we solve for distances

[upbeat pond]

Hold upwhich dist are we trying to find

[snow grail]

Sailor to blue

[upbeat pond]

OH

Ok wait ill write out this next step

Wait

Is it even possible to find x if its three angles

[snow grail]

There’s the distance given from red to blue

[upbeat pond]

OH

thanks i didnt notice that

[snow grail]

No problem

[snow grail]

That only works for right triangles

[upbeat pond]

I thought that was soh cah toa ?

What should i do

[snow grail]

Do you know how to convert an isosceles triangle to a right triangle?

[upbeat pond]

Nope

Dude wth I just searched up my teachers answer and his was done so fast I took like 40 minutes

[snow grail]

Oh wait you can do that

I was going for pythagoras

[upbeat pond]

Oh

I think they're both right but it's better to use the teachers method

[snow grail]

Been a hot minute since I’ve done trig or geometry, sorry

[upbeat pond]

Lol its ok ur still really good at it

[snow grail]

I personally prefer right triangles

[upbeat pond]

Same

wait no

idk what i prefer

whyd i get a diff answer

[snow grail]

This is why I prefer splitting an isosceles triangle into two right triangles

[upbeat pond]

Ohhh thats smart

[snow grail]

Since the vertical line bisects the angle and the base, you get two identical right triangles

To solve the distance from blue to sailor, you would need twice the distance from blue to midpoint

[upbeat pond]

I see

[snow grail]

Which would be 2(20sin(20))

Remember, if you have any triangle with two equal angles OR two equal sides, you have an isosceles triangle, meaning that the other condition is true

[upbeat pond]

Okay ill remember

I have a question how'd he get 13.7?

Mb if u dont know my trig is rusty too

[snow grail]

I recall very little about non right triangle Pythagorean theorem

[upbeat pond]

Ah okay its fine

[snow grail]

Also the answer that you got is equivalent to the answer he got

He just fully evaluated it using a calculator

[upbeat pond]

When i typed mine in the calc i got 19.25

strange

Oh WAIT

im In radians mode

I gott the right answer now

[snow grail]

$\frac{20\sin{40}}{\sin{70}}=13.681$

[wraith dagger]

Helmfirth

[snow grail]

Yeah, radians and degrees can kill a test

[upbeat pond]

Ugh im screwed then

[snow grail]

Don’t worry, just pay attention to the mode your calculator is in

[upbeat pond]

Theres a specific question that involves switching to radian mode

[snow grail]

I had an intro modern physics test where we had to constantly switch between radians and degrees

[upbeat pond]

Jeez they're setting u up for failure

Are u good at physics

[snow grail]

I actually got a 96 on it so I’m pretty happy with it

I’d say I’m pretty good at physics

[upbeat pond]

Wow for physics thats crazy

[snow grail]

It was the final, so I’ll have to live with not knowing what I didn’t know

[upbeat pond]

FINAL ?

Worth how much percent ?

[snow grail]

I don’t recall exactly how much but our professor let us take it open notes

[upbeat pond]

I doubt open note even helped

[snow grail]

He was a cool guy

Well, the formula sheet I had helped much more than my notes did

[upbeat pond]

What was the topic the test was on i wanna google it and see how difficult it looks

[snow grail]

So the class name was Intro Modern Physics

Want to take a guess?

[upbeat pond]

Intro Modern Physics

[snow grail]

Special relativity, quantum mechanics, Schrödinger’s wave equation, and hydrogen wave functions

[snow grail]

You had your finger on the trigger with that image

[upbeat pond]

Idk cause looking at that image makes me want to cry

[snow grail]

Don’t worry, I can explain them

[upbeat pond]

Please dont

I dont even want to try

[snow grail]

The first one describes how time passes for you when you move

[upbeat pond]

Im confused already

How is v c and 1/2 and a triangle telling me how time passes for me

[snow grail]

There’s no triangles here

[upbeat pond]

DELTA I mean

[snow grail]

Just physics trying to make sure you don’t go the speed of light

Oh delta just indicates a change in time

So t2-t1

[upbeat pond]

what does the t mean

[snow grail]

Delta t0 is the time that passes in the reference frame that is not moving, and delta t is the time that passes for the person that is moving

[upbeat pond]

Ok i didnt take physics for a reason

[snow grail]

This stuff is hard

Don’t worry too much about it

But just wait until you look at Schrödinger’s wave equation

[upbeat pond]

WHAT THE HELL 😭 😭

[snow grail]

Oh that’s not the whole thing

[upbeat pond]

they ran out of letters so they started using satans wand

Physics is the devil

[snow grail]

https://qph.cf2.quoracdn.net/main-qimg-002c4c1e9897063ed8459f4195e185a3

[upbeat pond]

WHAT 😭 😭

[snow grail]

The way to solve this is using differential equations, which is basically algebra except you use calculus instead of arithmetic

[upbeat pond]

What is that trident letter

Please i could barely figure out what a bearing was how am i supposed to know what ψ is

[snow grail]

Psi

It’s the actual wave function

[upbeat pond]

Oh

[snow grail]

Basically, you use the wave equation and methods of solving second order differential equations to solve for the wave function

[upbeat pond]

🧍🏻‍♀️

Do you know how to code?

[snow grail]

A little bit of python and a touch of matlab, and I’m learning LaTeX just to use the TeXit bot

[upbeat pond]

I aspire to have ur intelligence .

[snow grail]

It all comes down to loving math

If you can’t get enough of it, you’ll get farther in math then you ever thought possible

[upbeat pond]

What if i like math but i process information at a slower speed than everyone else

[snow grail]

Practice, and find ways to think about it so that you can solve it quickly

[upbeat pond]

Were u slow when U were younger or were u always quick to understand ?

[snow grail]

Admittedly I always tend to learn quickly

[upbeat pond]

Do u know other languages?

[snow grail]

I took Spanish in high school but I’ve fallen out of practice

[upbeat pond]

Oh i see

[snow grail]

What math are you in now btw

Like what’s the class called

Is that high school or college?

[upbeat pond]

Highschool

[snow grail]

Does your high school have an AP program?

Or IB?

[upbeat pond]

It has ib

I didnt go into ib though

[snow grail]

Fair enough, it is hard work

[upbeat pond]

Im an average student lol

[snow grail]

Are you interested in more complex mathematics?

[upbeat pond]

Idk, i'm not that good at math, my average right now is 88 which isnt good for grade 11

[snow grail]

That’s still an A

[upbeat pond]

Is it ?

[snow grail]

Yeah, I’m pretty sure you’re on the boundary of A and B

[upbeat pond]

Oh ok then its not bad

[snow grail]

Yeah, you’re doing just fine

Taken the SAT yet?

[upbeat pond]

Nope thats an american thing

[snow grail]

Ahhh

I shouldn’t have assumed

[upbeat pond]

its ok

[snow grail]

I was mostly looking for a standardized way to compare you to others other than making a guess based on a letter grade

But an A is still probably top 25 to 40 percent

Very rough estimate

[upbeat pond]

The class average in functions is 79%

Is 88 good compared to it?

[snow grail]

Although if you want to go into game design you will definitely need to learn calculus

[upbeat pond]

💔

[snow grail]

An 88 is good compared to a 79, but to know exactly how good, do you know the standard deviation of the scores

[upbeat pond]

I dont

I only know that 2 people failed

and everyone else is passing

[snow grail]

Keep it up then

[upbeat pond]

Thanks

[snow grail]

Don’t really have much more advice so I guess this channel can be closed

Anything else you need?

[upbeat pond]

I kinda do need help still

But Ill just see if u know this one question and if u dont thats fine

[snow grail]

Let’s do it

[upbeat pond]

A wheel with a radius of 1 metre picks up a nail on the ground. The wheel is traveling 108km/hour. What is the height of the nail at a time of 10 minutes after the wheel picks up the nail? Round your answer to the nearest tenth of a centimeter

[snow grail]

Ohhh now this is a fun one

I love weird questions like this

[upbeat pond]

All ik so far is that you draw a circle and then a graph with the parent function y = -cosx

[upbeat pond]

I dont ☝🏻

[snow grail]

The first step is to solve for the circumference of the circle

[upbeat pond]

Isnt it two pi r

[snow grail]

Yes

Next step is to find the speed in meters per second

[upbeat pond]

Okay so

108,000 m

?

[snow grail]

Yes

[upbeat pond]

Why do you convert it to meters though

OH

nvm i get it

[snow grail]

Next step is to convert 10 minutes to seconds

[upbeat pond]

600 seconds

[snow grail]

Now multiply the speed by the time

[upbeat pond]

64.8m

Jeez

[snow grail]

What did you get for speed?

[upbeat pond]

Wait

108,000 x 600 is it not ?

[snow grail]

That's 108000 meters per hour

[upbeat pond]

Ohh

......

[snow grail]

I would fear the wheel that travels at 108km/s

[upbeat pond]

i mean

[upbeat pond]

Fr wth

didnt even think about that

[snow grail]

So just divide by 60 twice to get m/s

[upbeat pond]

30

[snow grail]

Yes

Now multiply the speed by the time

[upbeat pond]

18000

[snow grail]

Perfect

Now, we assume this wheel has perfect traction

Divide the distance travelled by the circumference to find out how many rotations it did

[upbeat pond]

9000/pi ?

[snow grail]

Yes

Find an exact number, and include the decimal to as much precision as possible

[upbeat pond]

2864.788976

[snow grail]

Good

Now subtract 2864 to get just the decimal

This is the partial rotation at the end of the ten minutes

[upbeat pond]

genuine question how do you know this stuff 😭

[snow grail]

Honestly, I just kinda picture it in my head and then I figure out how to get there

My first thought was how to get to the end of the ten minutes, so first I converted everything to manageable units and then found the distance and circumference, divided to find rotations, subtracted for that partial rotation, and did the steps I'm about to say

Do you have just the decimal now

[upbeat pond]

Yes

[snow grail]

Multiply by 2pi to get how many radians that is

[upbeat pond]

4.957282411

[snow grail]

Perfect

Now substitute that into the equation y = 1 + sin(x - pi/2)

[upbeat pond]

Into x or y ?

[snow grail]

That maps the y value of the nail for the whole journey

Into x

Actually this is a good time to practice my latex

[upbeat pond]

I got 0.757547063 but idk if thats right

[snow grail]

$y=1+\sin(x-\frac{\pi}{2})$

[wraith dagger]

Helmfirth

[snow grail]

That's what I got

[upbeat pond]

Wowwo

[snow grail]

The way I got that equation is this:

1. The nail starts at y = 0, and never goes below that (or else it would be underground), which means the equation must be shifted 1 up
2. The nail starts at 0 and goes to 2, which means the sin function must start at -1, meaning the shift is minus pi/2 (This also could be -cosx but I was thinking of sin at the time)

I actually only just thought of the -cosx because I was thinking that cos is +pi/2, and the integral of sin is -cos, meaning the shift is in the opposite direction

Which is very inefficient but it works for me

[upbeat pond]

Yeah the teacher tells us to do the -cosx

I dont understand him but whatever

[snow grail]

Yeah, wheel height functions will be y = r - rcosx

Also you need to round the answer

I just looked back at the question

Nearest 10th of a centimeter, or nearest millimeter

so just round to 3 decimal places and divide by 100

[upbeat pond]

Okay

[snow grail]

Fun question though

[upbeat pond]

In what world

[snow grail]

Mine

I live for the weird questions

[upbeat pond]

I live for the straightforward questions

Unless u count this as straightforward

[snow grail]

It isn't really that straightforward

You need to go from distance to rotations to radians to height

[upbeat pond]

I feel like its maybe not that bad

I just forgot my conversions

or i dont know where to convert at what time

[snow grail]

It's fine, always good to practice

[upbeat pond]

True

[snow grail]

Anything else you need help with?

[upbeat pond]

Do u have the time? Cause damn

[snow grail]

I've got plenty of time

[upbeat pond]

Ok cause i'd rather not struggle on my own

Two roads intersect at an angle of 9 degrees. Bailey travels down one road at a speed of 20km/h, and Carmela travels down the other road at a speed of 32 km/h. Fifteen minutes later, a police helicopter above and between Bailey and Carmela is at an elevation of 500m and sees bailey at an angle of depression of 16 degrees. What is the straight-line distance from Carmela to the helicopter?

[snow grail]

This has a similar start to the nail wheel

First, convert the speeds to m/s and time to seconds

[upbeat pond]

Why do we do seconds

Do you have a problem with minutes

[snow grail]

I prefer working in seconds honestly, but yes we can work in minutes

[upbeat pond]

Oh okay

lets work in seconds anyways

[snow grail]

I mean you get the same answer

[upbeat pond]

Distance =speed x time

[snow grail]

Yes

[upbeat pond]

20 km /h x 15 min

[snow grail]

Also do you know if straight line distance is referring to the direct line from car to helicopter or the ground distance between them

[upbeat pond]

like this hold on

[snow grail]

Ah, ground distance

Do you have the distances?

[upbeat pond]

No immma be honest idk how to get them

[snow grail]

$d = (\frac{20 km}{h}*\frac{1 h}{60 min}*\frac{1 min}{60 s}*\frac{1000 m}{1 km})*(15 min*\frac{60 s}{1 min})$

[wraith dagger]

Helmfirth

[snow grail]

Those are all multiplied btw

Alternatively:

$d = (20 km/h)(15 min * \frac{1 h}{60 min})$

[wraith dagger]

Helmfirth

[upbeat pond]

Hmm'

Im gonna see how my teacher approached it

This is what he did

Is it the same thing ?

[snow grail]

Essentially, yes, he did it $d = (\frac{20 km}{1 h} * \frac{1 h}{60 min})(15 min)$

[wraith dagger]

Helmfirth

[snow grail]

Which is converting to km/min and then multiplying by minutes

Which is what you suggested a little while back

[upbeat pond]

Ohh

Sorry it takes a while for things in my brain to click

[snow grail]

No worries

As long as it does click, that's what matters

[upbeat pond]

True

Ok so the answer is 5km

[snow grail]

Yes

[upbeat pond]

Then for the other distance

32km/h x 15min

[snow grail]

Yep

Here's a little trick

15 mins = 1/4 hour

[upbeat pond]

8

8 for the other distance

[snow grail]

Yes

[upbeat pond]

Then at thispoint we have side angle side

[snow grail]

Let's work in km since it's a bit better with number length

Yes we do

I've made another right triangle because that

is what I'm used to

I sent that prematurely mb

[upbeat pond]

Its ok

[upbeat pond]

Dont teach me how u did this cause im going to use the method my teacher wants

Im gong to do

[snow grail]

Alright

We're going to find the distance between the cars

[upbeat pond]

x^2 = 5^2 + 8^2 - 2(5)(8)cos9

25 + 64 - 80cos9

89 - 80cos9

ngl idk what im doing now

hold on

[snow grail]

That's actually very very close to what I got, my answer's difference is likely due to rounding error

[upbeat pond]

What was ur answer?

[snow grail]

I got 3.160279734, I evaluated yours to be 3.159894421

[upbeat pond]

I just checked the teachers answer and its 3.16

So ur right and im close

[snow grail]

And that's what I rounded mine to

Yours is the exact answer

[upbeat pond]

Oh but its just rounded

[snow grail]

He got that and rounded to 2 decimal places

[upbeat pond]

Oh okay

[upbeat pond]

What do you do from this step cause my calculator gave me weird answers

[snow grail]

Is your calculator in radians

[upbeat pond]

Yeah

oh ugh

[snow grail]

No worries

Be prepared for the switch

[upbeat pond]

Im def gonna accidently be on radians the whole time just watch me

[snow grail]

In my high school I know some teachers gave partial credit for having the right method but the wrong calculator setting

[upbeat pond]

(during my test)

[upbeat pond]

Oh thats good to know atleast

[snow grail]

So now we need to find the distance from one car to the helicopter using the angle of depression

Remember that the helicopter's sightline has an angle of depression of 16 degrees and a height of 0.5 km

[upbeat pond]

hold on

[snow grail]

That works

I used the tree analogy from before but that does create a rectangle on both sides with those triangles as the complementary triangles

So use tan(16) to find the adjacent line

[upbeat pond]

0.28674

[snow grail]

How did you solve for that?

[upbeat pond]

I typed tan(16) into my calculator 😓

[snow grail]

Ah I see

Do you know what tan(x) is equal to in a triangle?

[upbeat pond]

the hypotenuse

[snow grail]

$\tan(\theta) = \frac{opposite}{adjacent}$

[wraith dagger]

Helmfirth

[snow grail]

I'm getting the hang of this

[upbeat pond]

Well opposite is 0.5 right

Wait

y/sin74 = 0.5/sin16

thats what my t eacher put

[snow grail]

Man your teacher hates right triangles

[upbeat pond]

LOL

[snow grail]

Oh wait that is a right triangle lmao

It's just the sine of the two angles under the opposite side, that's equal to the hypotenuse

sin(74) = y/hyp, hyp = y/sin(74)
sin(16) = 0.5/hyp, hyp = 0.5/sin(16)
y/sin(74) = 0.5/sin(16)

That is definitely one way to do it

Tangent is much easier

[upbeat pond]

Yeah

[tight hull]

I just do sohcahtoa

[snow grail]

That is what happened

[tight hull]

Sin=opposite/hypotenuse

[snow grail]

So once you solve for the distance to Bailey, you need to subtract that from the total distance between the two cars and then you have your answer

[upbeat pond]

Ill try right now

Ugh nvm I keep getting lost for no reason

Im gonna go to sleep

[snow grail]

Alright then, good night

Should this help thread be closed?

[upbeat pond]

Yes

Can u tell me what the answer is tho

Just so i can check tomorrow

[snow grail]

3.16-1.74=1.42

[upbeat pond]

ARE YOU SERIOUS

[snow grail]

Yeah

[upbeat pond]

No way i was confused over that

Ok anyways thank you so much for helping me

[snow grail]

No problem

Good luck and good night

[upbeat pond]

Thank you ❤️

.close

[cedar kiln]

Channel closed

Closed by @upbeat pond

Use .reopen if this was a mistake.

[cedar kiln]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[cloud birch]

i got 1.28/4 down but i forgot how to simplify the negative number

[cedar kiln]

@cloud birch Has your question been resolved?

[cloud birch]

wait nvm i found out ☠️☠️

.close

[cedar kiln]

Channel closed

Closed by @cloud birch

Use .reopen if this was a mistake.

[cedar kiln]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[unique ravine]

Hi i have a question:

[unique ravine]

lg = log_2 right?

and then how would i even solve for the inverse for this

do i distribute first or?

cause i switched the variables already n stuff

and i subtracted the 1

but then what?

[lusty grotto]

take y = 1 - lg(-2x)

and solve for x in terms of y

[unique ravine]

i have that part

[lusty grotto]

thats the inverse

[unique ravine]

ik

do i distribute the lg to the -2x?

[lusty grotto]

so?

[unique ravine]

cause im getting the inverse currently as

[lusty grotto]

u cud if u wanted to

what exactly di u mean by distribute

[unique ravine]

-1/lg(-2x) if i dont right?

[unique ravine]

like uh

log_2(-2) + log_2(x) kinda..

[lusty grotto]

log(ab) = log a + log b?

[unique ravine]

idk if its possible

yeah

[lusty grotto]

well u can write log(-2x) = log (2 * (-x))

[lusty grotto]

i dont get this

btw log_2(-2) does not make sense coz negative numbers dont have logarithms

[unique ravine]

oh

[lusty grotto]

ah right i get what u r saying

[unique ravine]

wait is log_2 = ln?

[lusty grotto]

ln is log_e

[unique ravine]

...

waitt i mean lg

log_2 = lg?

[lusty grotto]

idk what lg means, some use it for log_10

[unique ravine]

oh

[lusty grotto]

what did u learn

[unique ravine]

my teacher said log_2

but ive been searching

[lusty grotto]

okay then use that

[unique ravine]

and nothing shows up as lg

[lusty grotto]

right if ur tchr said log_2, use that

[unique ravine]

ok

so the inverse would be just

[lusty grotto]

so u wrote log_2(-2) + log_2(x) this is not really valid coz log_2(-2) is not defined

[unique ravine]

y=-1/-lg(-2x)?

[lusty grotto]

can u "distribute" tihs

[unique ravine]

i dont think so

[lusty grotto]

its the same thing u just did

[unique ravine]

oh

[lusty grotto]

so log(some number * another number) = log(some number) + log(another number)

when u first distributed, u took some number = -2, another number = x

now im asking u to take some number = 2, another number = -x

[unique ravine]

its the same thing tho

just different negative signs

[lusty grotto]

but now u have log_2(2) + log_2(-x) right

[unique ravine]

where did the negative go tho

oh

uh

[lusty grotto]

the log_2(-x) does make sense for negative x

[unique ravine]

not really

[lusty grotto]

for example if x = -10, then log_2(-x) = log_2[-(-10)] = log_2(10)

but if x is positive, then it wont make sense

btw u dont have to do this "distribution" to find the inverse

[unique ravine]

then how else would i find it..

cause im getting the wrong answer every time i do it

[lusty grotto]

so y = 1 - lg(-2x)

move all things with x to one side

what do u get

[unique ravine]

i got uh

-1/-lg(-2x)

[lusty grotto]

what

how did u get that

[unique ravine]

wait

lg(-2x)

dont we divide that

-1 + lg(-2x)?

[lusty grotto]

no treat the entire lg(-2x) thing as a variable

and subject it

in terms of y

[unique ravine]

^ so like this

[lusty grotto]

lg(-2x) = what

[unique ravine]

oh wait

x-1 = lg(-2y)

and then uh,

(x-1)/lg(-2)..?

[lusty grotto]

so we started with y = 1 - lg(-2x)

[unique ravine]

mhm

[lusty grotto]

and i ask lg(-2x) = what?

why did u put y inside lg

[unique ravine]

because when finding inverse, dont we switch variables

[lusty grotto]

not yet

[unique ravine]

oh

[lusty grotto]

so what is lg(-2x) = ?

[unique ravine]

idrk..

[lusty grotto]

so if y = 1 - x, what is x?

in terms of y?

forget the lg thing for a while

[unique ravine]

oh

so

[lusty grotto]

if y = 1 -x, then x = ?

[unique ravine]

so wait

we have to solve

for x first? from

[lusty grotto]

yes solve for x

[unique ravine]

lg(-2x)

[lusty grotto]

yeah

yeah

[unique ravine]

but why..?

if we're finding inverse

[lusty grotto]

the inverse tells u what value u must input to the function to get a certain output

[unique ravine]

oh ok

[lusty grotto]

so suppose when we input some value x, we get the output 5

[unique ravine]

yeah

[lusty grotto]

so 5 = 1 - lg(-2x)

u must solve for x to find what u input right?

[unique ravine]

yeah

[lusty grotto]

yeah

[unique ravine]

ok so uhm

-lg(-2x) = 0

[lusty grotto]

what is this

[unique ravine]

wait but

-lg = -log_2

so doesnt -2 cancel out?

[lusty grotto]

no

[unique ravine]

im so confused..

[lusty grotto]

okay forget lg for a while

[unique ravine]

ok

[lusty grotto]

say we had f(x) = 2x + 5

hw would u find the inverse

[unique ravine]

(x-5)/2 = y

is the inverse

[lusty grotto]

yeah

and how do u get this answer

[unique ravine]

i subtracted then divided

[lusty grotto]

do u know what an inverse is

like what it means

[unique ravine]

not really..

[lusty grotto]

why did u do this

okay

so if f(x) = 2x + 5

what u can do is

u can input various values, and find outputs

[unique ravine]

yeah

[lusty grotto]

maybe u input 2, then it outputs f(2) = 2*2 + 5 = 9

[unique ravine]

yes

[lusty grotto]

maybe u input -1, and it outputs f(-1) = 2*-1 + 5 = 3

etc

[unique ravine]

mhm

[lusty grotto]

lets say somebody asks u "what must u input, if u want the output to be 10?"

how would u answer this

[unique ravine]

then why dont we plug in 10 as the "y" kinda

like uh

10 = 2x+5

[lusty grotto]

yeah

and then

[unique ravine]

then solve from there

10-5/2

[lusty grotto]

yeah we solve for x

[unique ravine]

5/2

yeah

[lusty grotto]

exactly

so thats what we must input to get 10 as output

if u wanted to find out what u must input to get 100 as the output

[unique ravine]

yes

[lusty grotto]

u do the same thing right

[unique ravine]

mhm

[lusty grotto]

now lets say i am very interested in finding inputs which produce various given outputs

so i am going to define a function

[unique ravine]

ok

[lusty grotto]

lets call it g

now to this function i can input any number, and its output tells me what i shud input into f, to get that number as the output

do u understand that

[unique ravine]

yeah i understand that

[lusty grotto]

so maybe, when i input 10 into g, it outputs 5/2

?

get it?

[unique ravine]

yes

[lusty grotto]

yeah that sort of function is called an inverse

so g is the inverse of f

its a functino which tells u what values u must input to a certain function, to get a given output

[unique ravine]

yeah

oh

[lusty grotto]

okay now i want to find a formula for g

[unique ravine]

oh

[lusty grotto]

so when u input something into g, we know that it must subtract 5 from that number, and then divide it by 2

and output that

[unique ravine]

yeah

[lusty grotto]

and we know this because thats what we must do if i asked u this question

[unique ravine]

ok..

but for the log situations

how do i figure out how to get x alone

[lusty grotto]

so if f(x) = 1 - lg(-2x)

when x = -10, f outputs 1 - lg[-(-10)] = 1 - lg(10)

etc

maybe i ask u what must u input for x, so that the output is 10 say

how would u do that

[unique ravine]

i dont know..

[lusty grotto]

u must do exactly what u did when i asked u abt the 2x + 5 thing

[unique ravine]

ok so it would be like

uh

wait output as 10?

so 10 = 1-lg(-2x)

[lusty grotto]

yeah i guess thatll be less confusing

[unique ravine]

and what we input?

[lusty grotto]

exctly

[unique ravine]

to get that?

[lusty grotto]

yeah

[unique ravine]

how do i even solve for that tho

[lusty grotto]

what must x be so that u get 10 as the output

[lusty grotto]

so move all stuf with x to one side

and all other stuff to the other side

[unique ravine]

so

-9 = -lg(-2x)

[lusty grotto]

yeah

and u can also get rid of the negtive sign

[unique ravine]

oh right

[lusty grotto]

so lg(-2x) = 9 right?

[unique ravine]

yes

[lusty grotto]

right so now do u know what log means?

[unique ravine]

yes i think

[lusty grotto]

so log_2 (8) means "what number must u raise 2 to,to get 8"

[unique ravine]

3

[lusty grotto]

yeah

somebody tells u log_2 (x) = 5

what is x

[unique ravine]

5/log_2..?

but thats not possible..

[lusty grotto]

so the answer to "what number must u raise 2 to, to get x", is 5

is that clear

[unique ravine]

yes

[lusty grotto]

so then u must raise 2 to the power of 5 to get x

so what is x

[unique ravine]

uh..

oh wait

32?

[lusty grotto]

yeah

so do u get it

[unique ravine]

yes

[lusty grotto]

so here we have lg(-2x) = 9

if we take lg as log_2 coz ur teacher says that

log_2(-2x) = 9

so the answer to "what number must u raise 2 to, to get -2x"

is 9

9 is the number u must raise 2 to, to get -2x

so what is -2x?

[unique ravine]

-4.5?

[lusty grotto]

what

[unique ravine]

wait

[lusty grotto]

read that carefully

[unique ravine]

i dont really understand what that means

9^2=-2x

like that?

[lusty grotto]

no 9 is the number u must raise TWO to

raise TWO to the power of nine

do u get it

[unique ravine]

oh

[lusty grotto]

are u sure u get it

[unique ravine]

uhm i think..

but how does that = -2x?

[lusty grotto]

because when u write log_2(-2x) = 9

u have -2x inside brackets

[unique ravine]

oh

oh wait

so for rn

2^9 = -2x

[lusty grotto]

yeah

[unique ravine]

ok..

[lusty grotto]

so the log notation is like this, if u write
log_B(C)

this asks the question "what is the power u must raise B to, to get the number inside brackets"

do u get it

[unique ravine]

mhm

[lusty grotto]

u sure?

[unique ravine]

yes

i get it now

[lusty grotto]

okay then

so then what is x

u know what -2x is

[unique ravine]

yes

[lusty grotto]

-2x = 2^9

[unique ravine]

ye

so it would be

[lusty grotto]

so x is?

[unique ravine]

512 = -2x

512/2 = x

[lusty grotto]

yeah

[lusty grotto]

-x

u missed the -ve sign

[unique ravine]

oh right

[lusty grotto]

okay

so this was for when i asked what u must input to get 10 as the output

[unique ravine]

mhm

[lusty grotto]

so u must input -512/2 = -256

what if i asked what u must input to get 20 as the output

[unique ravine]

it would be 2^19 = -2x

[lusty grotto]

right

so x (which stands for what u must input) is then -(2^19)/2

right?

[unique ravine]

wait what

oh yes

[lusty grotto]

okay right?

[unique ravine]

ywa

yea

[lusty grotto]

okay so now u want to define a function which does this for u

call it g for now

u want to be able to input a number to g

[unique ravine]

ok

[lusty grotto]

and the output must be te number u must input to f, to get that number as output from f

so what must this functino be doing

so for example, g must output -256 when u input 10

[unique ravine]

yeah

[lusty grotto]

how did u arrive at -256 in terms of 10

[unique ravine]

basically inverse

[lusty grotto]

g must output -(2^19)/2, when u input 20

[unique ravine]

yeah

[lusty grotto]

how did u arrive at -(2^19)/2

like what calculations did u do

[unique ravine]

dividing -2

[lusty grotto]

what else

[unique ravine]

oh subtracting 1

from 20

[lusty grotto]

u input 20, g somhow turns it into -(2^19)/2

[lusty grotto]

exactly

so if u input a number y, to g

what must it do to y

[unique ravine]

subtract and divide?

[lusty grotto]

yeah subtract what divide by what

and do what else

[unique ravine]

1

[lusty grotto]

thats not all

theres something else u must do

[unique ravine]

the log thing

the log rule

[lusty grotto]

do u see a log thing here

[unique ravine]

no

[lusty grotto]

so why did u say that

[unique ravine]

because we used it

[lusty grotto]

g is not doing any log thing

[unique ravine]

before

[lusty grotto]

to 20

[lusty grotto]

yeah what did we do to get rid of the log thing

[unique ravine]

mhm

[lusty grotto]

?

[unique ravine]

we used the rule right?

its like

log_2 = 19

which means

2^19 =

[lusty grotto]

so again, g outputs -256 when u input 10, and -256 is just -(2^9)/2

it outputs -(2^19)/2, when u input 20

do u see a pattern

[lusty grotto]

yeah

[unique ravine]

yeah

i see it

[lusty grotto]

so if u input y

what must it output

[unique ravine]

g?

[lusty grotto]

for 20 u get -(2^19)/2
for 10 u get -(2^9)/2
for y u get?

[lusty grotto]

yeah

[unique ravine]

so

[lusty grotto]

can u answer this

[unique ravine]

o

for y i get

-(-2^y-1)/2

[lusty grotto]

exactly

so thats it

thats the invers

[unique ravine]

ohh ok

[lusty grotto]

so g(y) = -(2^(y-1))/2

right

[unique ravine]

yes

[lusty grotto]

now theres some specific notation we use to denote the inverse

we dont call it g, we call it $f^{-1}$

[wraith dagger]

SilverSoldier

[unique ravine]

mhm

[lusty grotto]

so $f^{-1}(y) = -\dfrac{2^{y-1}}{2}$

[wraith dagger]

SilverSoldier

[lusty grotto]

now u can just replace the y with x

[unique ravine]

oh ok

[lusty grotto]

and wite $f^{-1}(x) = -\frac{2^{x-1}}{2}$

[wraith dagger]

SilverSoldier

[lusty grotto]

are u sure u understood this

[unique ravine]

yes i understood that much

[lusty grotto]

so just to see if u got it, if u had f(x) = log(1/x) + x

how might u find the inverse?

say log means log_2

[unique ravine]

ok so

[lusty grotto]

uhm

wait this might be difficult

[unique ravine]

wait +x

?

[lusty grotto]

dont do this

[unique ravine]

ok

uhm..

[lusty grotto]

uhm

say u had f(x) = log[x/x+1] - 10

find its inverse

[unique ravine]

ok so first

[lusty grotto]

$f(x)=\log\left(\dfrac{x}{x+1}\right)-10$

[wraith dagger]

SilverSoldier

[unique ravine]

10 = log(x/x+1)

wait is log_2?

or just log?

[lusty grotto]

uhmmm

[unique ravine]

is log = 10?

[lusty grotto]

why did u take 10 = log(x/x+1)

[unique ravine]

because i subtracted 10

[lusty grotto]

idk take it as 2

[unique ravine]

from the left

[unique ravine]

ok

[lusty grotto]

what is the left

[unique ravine]

i mean from the right

i added 10

from the right side

of the equation

[lusty grotto]

what did u add it to

[unique ravine]

y + 10

[lusty grotto]

okay so keep the y

[unique ravine]

ok

[lusty grotto]

y+10 = log(x/x+1)

now?

[unique ravine]

2^y+10 = x/x+1

[lusty grotto]

yeah

2^(y+10) = x/x+1

[unique ravine]

now we uhm

[lusty grotto]

it 2^(y+10) is a lot to type, just write t for it instead

so u have t = x/x+1

[unique ravine]

ok

[lusty grotto]

then?

[unique ravine]

then uh

do we do t*(x+1)?

[lusty grotto]

yeah u can do it

[unique ravine]

ok

so t * (x+1) = x?

[lusty grotto]

yeah

then

[unique ravine]

then we switch the signs?

[lusty grotto]

what

[unique ravine]

idk what to do after..

[lusty grotto]

u r supposed to find x

[unique ravine]

oh wait

wait how do i do that.. if theres two x's

[lusty grotto]

so i give u this, ur supposed to come up wth a function that can find what u must input into this to get a give number as output

[lusty grotto]

mutliply thru by t

in the t * (x+1)

so i might have to go now

dm me if u want or maybe somebody else will help you

[unique ravine]

ok

[lusty grotto]

ill be back in idk an hour or half hour

[unique ravine]

👍

.close

[cedar kiln]

Channel closed

Closed by @unique ravine

Use .reopen if this was a mistake.

[unique ravine]

.reopen

[cedar kiln]

✅

[unique ravine]

can someone help me understand this

i think ive got the inverse down with silvers help

but how do i move on from after that

[crimson sedge]

Hi @unique ravine

[unique ravine]

hi

[crimson sedge]

Sooo what do you know about the domain and range between the inverse and its function

Also, what was your inverse

Also is that base 10?

[unique ravine]

base 2

inverse is currently -(2^(x-1))/2

[crimson sedge]

Okay doesn't matter for this tbh

[unique ravine]

o

ok

[crimson sedge]

Well anyways the domain of a log is always x > 0

[unique ravine]

so horizontal asymptote at 0?

[crimson sedge]

But since your log has a negative in it, the thing needs to be in reverse

Like if you have x = 2 you get log(-4) which is undefined

So your defined points are when that thing is positive

Soo what makes a negative positive?

[crimson sedge]

Not horizontal

[unique ravine]

i lost u on x = 2 i get log(-4)

oh wait

if u plug into original

function

then thats log_2(-4)

[crimson sedge]

Yeah

[unique ravine]

which isnt possible

[crimson sedge]

Yes

Exactly

[unique ravine]

so id need a negative number

to make it positive

[crimson sedge]

Yes!

Exactly

[unique ravine]

that makes sense

[crimson sedge]

So that domain is just x < 0 in this case

[unique ravine]

mhm

so (0, infinity)

[crimson sedge]

Nope

[unique ravine]

oh

[crimson sedge]

That would mean x > 0

[unique ravine]

wait

oh wait

ur right

[crimson sedge]

Yee

[unique ravine]

sorry its like 3 am and im just trying to understand this stuff :')

but yes

(-infinity, 0)

[crimson sedge]

Lmaoo its okay dw

[crimson sedge]

But yeah that's right

Now for the range

Actually log(-2x) is more or less the same thing as say log(2x)

[unique ravine]

ok

[crimson sedge]

The difference is that it is "reflected" And accepts negative values only

[unique ravine]

mhm

[crimson sedge]

So what is the range of a normal log function?

[unique ravine]

isnt it -infinity, infinity?

[crimson sedge]

Yeeee

It is the same thing here because

You can input any number you would normally have in a normal log function

[unique ravine]

yeah

[crimson sedge]

Like you can have log(100000), log(0.001), etc. As long as you are multiplying it appropriately with the negative x domain we talked about

[unique ravine]

so when does the range ever change?

[crimson sedge]

In this case you are just reflecting the thing across the y axis so it will be the same range, but it will be different if you had say, extra conditions you needed to put on it

[unique ravine]

o

conditions like what?

actually i dont think hes going to go over that

[crimson sedge]

Like if you have something crazy like idk

[
\sqrt{\log(x)}
]

[wraith dagger]

♡LexQa♡

[unique ravine]

o

[crimson sedge]

I don't think the range of a log changes unless you are doing something to the whole function like taking the square root of the entire thing

[unique ravine]

ok

[crimson sedge]

Or raising the whole thing to a power and so on

[unique ravine]

mhm

ok so uhm

[crimson sedge]

Okay so that's cool and all, but also

[unique ravine]

so domain and range is there for f(x)

then for inverse

uh

my -2^x-1/2

isnt the domain supposedly 2, infinity?

i have no clue how this works..

[crimson sedge]

Here is what I was going to get into

So actually, if you have a one-to-one function, and you find the inverse of that, the domain and range of the function interchange for the inverse

[unique ravine]

oh

oh wait rly

so that would mean

the range is (-infinity, 0)

while domain is (-infinity, infinity) ?

[crimson sedge]

So basically:
[
f(x) = 1- \log(-2x) \quad D: (-\infty, 0), R: (-\infty, \infty)
]
[
f^{-1}(x) = -2^{x-2} \quad D: (-\infty, \infty) R : (-\infty, 0)
]

[unique ravine]

wait

i have a dif inverse

[wraith dagger]

♡LexQa♡

[crimson sedge]

It's the same

I simplified it

[unique ravine]

wait hows that simplified tho

cus isn't it over 2

[crimson sedge]

[
\frac{-2^{x-1}}{2} = -2^{x-2}
]

[wraith dagger]

♡LexQa♡

[unique ravine]

oh

[crimson sedge]

It's just

Exponents law

Like

[unique ravine]

o

[crimson sedge]

[
\frac{a^n}{a^m} = a^{n-m}
]

[wraith dagger]

♡LexQa♡

[unique ravine]

oh ur right

wait wut

isnt the base 2?

[crimson sedge]

Yeee and actually, u couldve found the domain and range of our function using that procedure

As well

[crimson sedge]

Yes

[unique ravine]

actually i rly never knew that it was just switched so ty

so when i graph this

[crimson sedge]

Oh but like I said

This is only for one-to-one functions

[unique ravine]

wait one -to - one?

[crimson sedge]

For not one to one functions the inverse won't have the same domain and range, you have to restrict the domain

[crimson sedge]

Uhh have u ever heard of the horizontal line test

[unique ravine]

yes

oh so basically not like x^3

kinda

or x^2

[crimson sedge]

U mean x^2 but yeah

[unique ravine]

yea

[crimson sedge]

Because x^2 ain't injective

[unique ravine]

mhm

[crimson sedge]

Are u confused by anything doe, I can explain in a different way if u r wondering about something

[unique ravine]

i think i understood it more now

im gonna keep working on it

[crimson sedge]

Ayeee that's amazing

[unique ravine]

but u said a reflection right