help-38

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[azure barn]

Can someone help me with quadratics

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[azure barn]

<@&286206848099549185>

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[sonic river]

wsg

[azure barn]

please help me

[wraith hinge]

send your question here

[azure barn]

ok

[wraith hinge]

i will be able to help in quads

[azure barn]

Ok ty

A and b

[wraith hinge]

,rotate

[wraith hinge]

have you heard of the quadratic formula

[azure barn]

Yes

I have

I don’t know how to plug it in tho

[wraith hinge]

$x = \frac{-b \pm \sqrt{b^{2}-4ac}}}{2a}

[azure barn]

and what to do with the equal 0

[wraith hinge]

jesus i cant teach without texit

oh ty

。。。

[azure barn]

Uh

[wraith hinge]

$x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}$

[solid kiln]

jl7up~

[wraith hinge]

ty

[wraith hinge]

where
a is the coefficient of x^2
b is the coefficient of x
c is the constant

@azure barn

[azure barn]

Ok

@wraith hinge

Then what

It’s equal to zero

What do I do about that

Uh

someone pls help me

[tulip violet]

That equals to 0 just means it's an equation so you solve for x

You just plug the values into the quadratic formula and simplify

[azure barn]

Hold up

Can u stay with me

And tell me if I’m correct

[wraith hinge]

sorry was off to edit the paralympics wikipedia 😂

thought pings overtook mutes

[azure barn]

it’s ok

I think I’m doing this wrong

[tulip violet]

Show your work

[tulip violet]

,rotate

[tulip violet]

That's correct, why do you think it's wrong?

[azure barn]

What do I do now

What’s the square root of 73

[tulip violet]

Well, it's x = that stuff

[azure barn]

huh

[tulip violet]

The goal is to solve for x

[azure barn]

Yes

[tulip violet]

So you should denote that it's x = ....

All you wrote was a fraction

[azure barn]

Shouldn’t there be 2 final answers

[tulip violet]

There is

That's what the $\pm$ means

[solid kiln]

CaptainNova22

[azure barn]

okay

But like

There isn’t a perfect root for 73

[tulip violet]

That's true

[azure barn]

then what

[tulip violet]

There's nothing more you can do

You just leave it like that

[azure barn]

I get a super long decimal

Oh okay

So what I wrote is my final answer??

what about at it’s simplest form

[tulip violet]

Can you simplify that fraction?

[azure barn]

Idk

Can I

[tulip violet]

That's what I'm asking you

[azure barn]

I don’t think so 😭

[tulip violet]

Do you know how to simplify fractions and radicals like that?

[azure barn]

Nope

[tulip violet]

I suggesting viewing this as an example https://www.youtube.com/watch?v=3ayhvAI3IeY

[azure barn]

Oh okay

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[west light]

Hello

[west light]

I need to find the angles of a triangle and have given sides

Must use cos

The crosse out one for the angles

[split chasm]

what issue did you have when applying that ?

[west light]

Just want to know what order to put them in I have a tendency to double check a lot

That’s how I put it in but I know I have to do it again because you need 2 angles

[split chasm]

looks like you messed up signs or plugged in the wrong values

from labelling conventions,
Capital vertex/angle is opposite the lowercase side

how did you end up with
9^2 -15^2 +8^2
on the numerator

[west light]

Yeah I must’ve done something wrong because I did another one and they were both over 100 and it’s supposed to add up to 180 but I put them in the numerator because I thought that’s how you found the angles

[split chasm]

first state what formula you have before plugging in your values

what you're applying seems to be a bit different from what you had earlier

[west light]

Maybe 15 was supposed be first and I think I switched the plus and minus sign based on other examples

[west light]

It’s the law of cosine

For angles

[split chasm]

like actually write it out

when you stated the law earlier, had a negative denominator
show what you did to get the law with a positive denominator

[west light]

I switched to this one I think my teacher crossed the other one out for a reason probably

[split chasm]

now note that here, you're supposed to subtract (side opposite)^2

(and add the square of the other two sides)

[west light]

Subtract what?

[west light]

Ok I tried fixing it but I think there’s stull something wrong

[split chasm]

,rotate

[west light]

Ignore the 107 I didn’t change that part yet

[split chasm]

those look alright

[west light]

Are you sure?

So the last angle is 123?

[split chasm]

about that, but not rounded correctly

[west light]

I need to round to the hundredths

[split chasm]

that would mean to 2dp, so you'd need to go back and round the values you got earlier again

[west light]

But if the setups good shouldn’t I have gotten 50-70 numbers

[split chasm]

wdym 50-70 numbers

[west light]

The angles would’ve been 50 to 70 degrees

[split chasm]

why are you making that assumption

[west light]

That’s what happened in the previous problem

[split chasm]

you'd only expect that if all the side lengths are quite close together

[west light]

Okay

I’m doing delta math right now so maybe that’ll help me out with it idk

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[west light]

Close

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[brittle root]

hi

[brittle root]

i have to show this

where w and zi are complex numbers

could u pls pls tell me what is wrong with this

there should not be a "n" in denominator

<@&286206848099549185>

[zinc ginkgo]

Try redoing your proof for n=2

[brittle root]

ok i try it again ty

but is there any simpler way to show that?

[zinc ginkgo]

If there is, it'll appear in n=2 or 3

[brittle root]

alright tysm

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[brittle root]

.close

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[lofty isle]

anybody here has experience with data analysis ?

[wraith hinge]

send your question here and there will be someone to help you

I did in one channel

But no one online

[lofty isle]

I don't have a problem or smth I need some advice about the field only

[wraith hinge]

there exists a helpers role

in case you didnt notice and therefore didnt ping

[wraith hinge]

ah k

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[mint wadi]

!help

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[mint wadi]

Hello

Help

[meager bloom]

Well find out at the end of each contract how much salary will Christian be having

Select the one which is more

[mint wadi]

I solved it and they have the same salary

[meager bloom]

How did you calculate? Show me

I can clearly see difference without even calculating 😅

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[full dock]

how do i do this?

[digital gulch]

what is p

[full dock]

it is P(X=x)

[wispy sparrow]

what can you say about the sum of probabilities?

[full dock]

it equals 1

[wispy sparrow]

so a=?

[full dock]

um

a = 1

oh bet bet a = 1 g

.close

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[elfin gyro]

I'm not sure how to do this. So for I've separated 3x and 1

[wind cloak]

HINT: Divide the numerator and denominator by x

[elfin gyro]

wait but i can do that for the one under the radical?

[wind cloak]

erm, wdym?

[elfin gyro]

right the top would simplify to 3

bc of the infinity but how would i do the bottom

the denominator

?

[wind cloak]

$\sqrt{\frac{1}{x^{2}}}\sqrt{x^{2}+2}$

[solid kiln]

77²

[elfin gyro]

oh so i'd get 1

but wouldn't ihave to divide bt x^2 on the numerator too?

[wind cloak]

it is in a square root dude

[elfin gyro]

ohh

[wind cloak]

it is just a simple 1/x

[elfin gyro]

i see now

thanks

.close

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[modern heart]

How can I prove that $$\frac{2x+2xy-y^2}{2y+2xy-x^2} = \frac{1}{(1+x)^2} \frac{y^2}{y^2}\frac{-x}{x} // \text{if} // y^2(1+x) = x^2(1+y)$$ I've tried substituting but it had no help

[solid kiln]

Boraed

[modern heart]

I am stuck at this

$$
\frac{\frac{-2y\sqrt{1+x}}{\sqrt{1+y}} + 2xy - \frac{x^2(1+y)}{1+x}}{\frac{-2x\sqrt{1+y}}{\sqrt{1+x}} + 2xy - \frac{y^2(1+x)}{1+y}}
$$

[solid kiln]

Boraed

[modern heart]

I don't know how to simplify to what I need

[wind cloak]

what exactly the expression on right hand side is?

[pearl lotus]

2x + 2y +2xy +2 = 2(1+x) (1+y)
So try adding the terms not available and try factoring like this

[modern heart]

in short I have to show that LHS is equal to $$\frac{-1}{(1+x)^2}$

[solid kiln]

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

[wind cloak]

erm okok

[pearl lotus]

Also u can use the relation given

[modern heart]

it doesn't work

i've tried that

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[modern heart]

bump

[modern heart]

problem is still to prove $$\frac{2x+2xy-y^2}{2y+2xy-x^2} = \frac{-1}{(1+x)^2} // \text{if} // y^2(1+x) = x^2(1+y)$$

[solid kiln]

Boraed

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@modern heart Has your question been resolved?

[zinc ginkgo]

try solving for y^2 from the given y^2(1+x) = x^2(1+y) and plug it into the LHS of the first equation

[modern heart]

how will this help

this just gives the numerator as

$$ 2x +2xy + 3yx^2 + x^2$$

[solid kiln]

Boraed

[zinc ginkgo]

when solving for y^2=..., you'll get a factor of (1+x) in the denominator which is also on the right side of the first equation

[modern heart]

yes we get that

but when I put values of y^2 in the denominator, the 1+x cancels each other out

[zinc ginkgo]

the resulting denominator should be able to factor out another (1+x) and the rest should cancel

[modern heart]

$$(-1)\frac{2x +2xy + 3yx^2 + x^2}{(1+x)(x^2 - 2xy -2y)}$$

this is the equation which i get

if I try to solve for the quadratic in the denominator, (1+x) cancels out and I am left with the initial equation again

[zinc ginkgo]

i don't see how you get this

can you show your whole steps

[solid kiln]

Boraed

[modern heart]

$$\frac{2x+2xy - x^2\frac{1+y}{1+x}}{2y+2xy-x^2}$$ this further simplifies to
$$\frac{2x(1+x)+2xy(1+x) - x^2(1+y)}{({1+x})(2y+2xy-x^2)}$$ after taking (-) common from the denominator and multiplying terms in the numerator we get
$$(-1)\frac{2x +2xy + 3yx^2 + x^2}{({1+x})(2y+2xy-x^2)}$$

[solid kiln]

Boraed

[modern heart]

@zinc ginkgo even if try to force factor out (1+x) it still leaves +2 as a denominator.

$$([1+x)(x-2y-2)] +2$$

[solid kiln]

Boraed

[modern heart]

factoring numerator of the last expression gives $$(-1)\frac{(x^2 + 2x)(1+y)}{({1+x})(2y+2xy-x^2)}$$

[solid kiln]

Boraed

[modern heart]

it my bad I forgot to take account for the (-) i took common

[modern heart]

factoring numerator of the last expression gives $$(-1)\frac{(x^2 + 2x)(1+y)}{({1+x})(x^2-2xy-2y)}$$

[solid kiln]

Boraed

[modern heart]

this is the correct expression

[wraith hinge]

interesting

@modern heart so basically we know that xy = -(x+y)

[modern heart]

yes

[wraith hinge]

i used this btw

mhm

[modern heart]

shall i use compendo and dividendo or is that not needed

[modern heart]

where should I use this btw

in the denominator one?

[wraith hinge]

i would say both num and denom

we can clearly see the 2x getting cancelled right away

same in denom with 2y

[modern heart]

okay I used it in the denominator here

the terms cancelled out

but still

it is not giving the answer

[wraith hinge]

atleast we are somewhere better now

[modern heart]

$$(-1)\frac{1+y}{1+x}$$

this is what I have

[solid kiln]

Boraed

[wraith hinge]

i got (2y -y^2)/2x-x^2

[modern heart]

let me try this

[modern heart]

how?

[wraith hinge]

you got this result ?

[modern heart]

okay yeah i got that

i have what you have

[wraith hinge]

hmm now what lol

[modern heart]

i get a different result than you

i have plus(s) where you have minus(s)

[modern heart]

i put that here\

[wraith hinge]

oh yeah mb

(2y +y^2)/2x+x^2

[modern heart]

now what

we're stuck again

[wraith hinge]

lemme try a thing

nah nvm

you got solution ?

[modern heart]

nope

still stuck

I think that this is indeed unsolvable

i asked someone and they told me to simplify the expression in the form of $$\frac{1}{(1+x)^2} \cdot \frac{y^2}{y^2} \cdot \frac{-x}{x}$$

i can't do that

[solid kiln]

Boraed

[modern heart]

but how to do this

<@&286206848099549185>

[wraith hinge]

i think this is the most simplified version

without any sq roots

[modern heart]

they want an answer only in the terms of x

[wraith hinge]

is this the original question ?

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@modern heart Has your question been resolved?

[modern heart]

yes

[opaque jetty]

This might be a weird take, but what if we write -1/(1+x^2) = -y^4/x^4(1+y^2)

[opaque jetty]

Update: do not.

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[fallen lichen]

bro

[fallen lichen]

my teacher wants me to show my work

I DONT HAVE ENOUGH ROOM

YOU GIVE NOT EVEN 4 LINES AND I NEED LIKE 11

SHE SAID TO USE ANOTHER PAPER

LIKE WTF BRO ISTG

IF I CAN PROVE I KNOW THE WORK WHY DO I HAVE TO SHOW MY WORK

[bright quarry]

🤔

interesting place to rant

[fallen lichen]

yes

[bright quarry]

https://discord.com/channels/268882317391429632/268882317391429632

go here

[fallen lichen]

BUTS TILL BRO I SWEAR

wait

i need help

:(

[bright quarry]

then send the question

[fallen lichen]

okay so its basically rearranging variables

i dont have a specific question

but theres so many different types on rearrangments

and i took a quiz today and idk if i did it right

so i just wanna learn how to do it right

[bright quarry]

if you don’t have a specific question then go to khan academy

[fallen lichen]

.close

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[dusk summit]

.

[dusk summit]

am i right with C here?

[simple haven]

@dusk summit I could simply answer yes or no, but I'm curious why you think the answer is 4.

Did you guess?

[dusk summit]

i tried explaining it on my assignment

[simple haven]

You claim two real zeroes. What are they?

[dusk summit]

the roots 2+ 3i and 2 - 3i contribute two zeros to the polynomial

right

[simple haven]

Yes

But what are the two real zeroes?

In the table

Because I only see one 👀

[dusk summit]

oh -1

yeah true

wait so i messed up

[simple haven]

Finally, is this answer multiple choice? As in can you choose multiple options?

[dusk summit]

yes just one option

[simple haven]

Because there are two possible answers based on a pedantic reading

[dusk summit]

i think it's 3 now

B

[simple haven]

But likely they want 3

[dusk summit]

yeah thats what i was thinking now

[simple haven]

Technically, it didn't say that there wasn't another pair of complex roots

So it could be any odd number ≥ 3. But I think they want 3.

[dusk summit]

true you're rightt

[dusk summit]

same i'll just go with B i guess

[simple haven]

That's also what I would go with

[dusk summit]

thank you again

i have like 4 more i need to do

.close

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[dusk summit]

am i right with D here?

[wicked badge]

what do you think

[dusk summit]

it's either A or D

[wicked badge]

well thats one way to solve it lol

[dusk summit]

😭

[wicked badge]

anyways you got it lol

[dusk summit]

thank you man

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[swift forge]

help

[swift forge]

how can i know the domain of all of these

[strange raptor]

domain will be the values of x where the function exists

[swift forge]

so what’s the answer

i don’t know how to tell

16,0

[strange raptor]

plug in random values of x and see if it exists at that point

an example of a domain is ( -infinity , infinity )

[swift forge]

this can’t b infinity tho

it’s squareroot

[strange raptor]

okay good you're getting somewhere

[swift forge]

ok so how can i know the domain

[strange raptor]

if you know what x can't be

you know what x CAN be

[swift forge]

how can i find out what it can’t be

[strange raptor]

this

[swift forge]

so 16

?

[strange raptor]

plug in 16 for x

what do you get

[swift forge]

0

32

[strange raptor]

right

sqrt 32 exists

so

16 is part of the domain

[swift forge]

6

huh?

i could’ve plugged in like 80

[strange raptor]

yes

[swift forge]

and if the square roots for that works

that means it’s the domain?

[strange raptor]

no

[swift forge]

that makes close to no sense

[strange raptor]

its IN the domain

[swift forge]

wdym

[strange raptor]

( -infinity , infinity )

is the domain for ALL the numbers

but since we established that some of the numbers that dont work

how can you change the domain to only include numbers that work

[swift forge]

bro im lost

[strange raptor]

do you know what a domain is

[swift forge]

yes

it’s x

[strange raptor]

kinda

its all values of x that work for the function

[swift forge]

true

[swift forge]

yes

[strange raptor]

this is a valid function

so the domain contains 16

[swift forge]

yea

but it could go on forceer

forever

[strange raptor]

right

so

[swift forge]

they’re asking for the interval notation

[strange raptor]

( -infinity , infinity )

this

means forever

right

[swift forge]

yes

[strange raptor]

in both directions

but you just said

it can't go on forever

[swift forge]

it can

i said

but i can’t wright every single number

[strange raptor]

right here

here is your hint

[swift forge]

well i was wrong

[strange raptor]

infinity = every positive number

-infinity = every negative number

[swift forge]

so every positive number works in this situation

[strange raptor]

yes ill give you that

[swift forge]

or only the numbers that can get square rooted

what does it mean to give them the domain in interval notation

[strange raptor]

( 5, 8)

[swift forge]

cause that’s what they asking for

[strange raptor]

contains numbers between 5 and 8

( -infinity , infinity )

contains numbers between -infinity to infinity

[swift forge]

ok

ur point being?

[strange raptor]

i just told you what interval notation was

( f, g)

[swift forge]

numbers between 5 and 8

[strange raptor]

or (f, g) U (h, k)

is the form of interval notation

[swift forge]

bro i don’t get it

[strange raptor]

you can test negative numbers as well

[swift forge]

is there more then one answer

[strange raptor]

number line yes

you understand?

[swift forge]

i don’t understand the answer to the question

[strange raptor]

do you understand what the picture is trying to tell you about a number line

[swift forge]

yes

infinity

[strange raptor]

no

[swift forge]

oh

[strange raptor]

its showing you numbers between -10 and 10

[swift forge]

ok

[strange raptor]

yellow and red

[swift forge]

ok

[strange raptor]

two intervals

[swift forge]

got it

[strange raptor]

and then I wrote it in interval notation

so the questino is asking what interval

does x exist

[swift forge]

yes

[strange raptor]

to make the function valid

[swift forge]

yes

[strange raptor]

what is the lower bound

of the interval

so if it can't go on forever bc its square root

what CAN it go down to

what property of the function

limits how far X can go

[swift forge]

0

[strange raptor]

test it

x=0

what does that give you

[swift forge]

0+16

16

4

squareroot

[strange raptor]

so it exists

[swift forge]

so what doesn’t exist

[strange raptor]

0 is not the lower bound

go even lower

[swift forge]

-1 +16

15

cant

[strange raptor]

sqrt (15) is still a number

so it works

[swift forge]

it’s infinity

every number works

[strange raptor]

test it

[swift forge]

u want me to test every single number

[strange raptor]

yea

[swift forge]

-17+16

-1

squared

nope

[strange raptor]

squared or sqrt

anyways

can you take a sqrt of a negative number

[swift forge]

no

[strange raptor]

so x cant = a number that makes the function sqrt a negative number

that's your lower bound

[swift forge]

-17 is my lower bound

[strange raptor]

so x cant = a number that makes the function sqrt a negative number

your lower bound is whatever number right before that happens

[swift forge]

so -17

[strange raptor]

or -16.5

or -16.01

[swift forge]

yea

[strange raptor]

or -16.0000000001

[swift forge]

i get it

[strange raptor]

so in interval notation

we use round brackets

( )

[swift forge]

yep

[strange raptor]

to say all values BETWEEN the numbers

[swift forge]

ok

[strange raptor]

square brackets [ ]

to say values including those numbers

you can combine them

( ]

but yea

[swift forge]

why is it that some go like (c,c]

[strange raptor]

( f, g]

not including f but including g

[swift forge]

ok ok

so what’s the answer

[strange raptor]

you found your lower bound

bounds

and upper bound

[swift forge]

what do i do with it

[strange raptor]

put it into interval notation

[swift forge]

i didn’t find the upper bound

[strange raptor]

why not

[swift forge]

is it 16

[strange raptor]

16 is not the upper bound

[swift forge]

what is it

[strange raptor]

the highest X can be

[strange raptor]

just told you 16 is not the upper bound

https://www.youtube.com/watch?app=desktop&v=Ww7xtG2S7IM

ping me when you finish this video

he's a better teacher than me

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[little condor]

how to solve without using h'opital rule

[strange raptor]

https://tenor.com/view/jason-morgan-jason-and-sam-sam-mccall-gh-general-hospital-gif-11291343265192287077

[ionic pendant]

use these known limits

[little condor]

how i use them

ik how to do it for the sin

but for cos i dont get it

[somber lance]

Convert the 1- cos 6x = sin^2 3x

Then you can use the (sin x)/x formula

[somber lance]

.

[little condor]

how u convert

[austere cedar]

I wouldn't convert.

Note if you let θ = 6x or θ = 8x, you don't have the right form for your question.

You want to algebraically change the 8x into a 6x.

This is easy to do by multiplying and dividing by 6

[little condor]

oh

so 1-cos(6x)/6x = 0?

[little condor]

so like that?

[austere cedar]

Yeah that looks good!

[little condor]

ok thanks

.close

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[somber lance]

From cos2x= cos^2(x) -sin^2(x)

Just keep 3x instead of x

You should get that form

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[fair meteor]

what do they mean by this exactly

like whats number of leading ones

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[marsh forum]

I'm not sure I understand how the rotation is taking place here

[marsh forum]

So like the x axis rotates, rotating the y and z components

[broken pilot]

To find the matrix, you can find where the basis vectors end up after the transformation; these are the columns of the matrix

[marsh forum]

So the z components new position confuses me

oh wait

nvm

thanks

$\begin{bmatrix} 1&0&0\0&cos(t)&-sin(t)\0&sin(t)&cos(t)\end{bmatrix}$

[solid kiln]

Veni, vidi, perii

[marsh forum]

Got it!

How would I find a general rotation matrix for $\R^n$ though

[solid kiln]

Veni, vidi, perii

[marsh forum]

Let's say about a diagonal axis

in say $\R^4$

[solid kiln]

Veni, vidi, perii

[marsh forum]

probably beyond me for now

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[eager lynx]

Can someone explain this to me?

[worthy tree]

do u know what the definition of an inverse function is?

[eager lynx]

Yeah

swap y and x

I jsut dont know how to do domain/range

[worthy tree]

well it would help if u know the formal def

[eager lynx]

for example the invere for that equation is y=root x-3

[worthy tree]

the idea is that f(x) = (x+3)^2 can be a function for any domain that is the subset of the real numbers

[eager lynx]

I thought it cant be -3

because it cancels it out to 0

[worthy tree]

e.g. u can restrict the domain to the interval [0, 1]

[worthy tree]

what cant be -3?

[eager lynx]

this

acuse -3 cancels out 3

cause*

[worthy tree]

yeah f(-3) = 0?

so what?

[eager lynx]

wait am I buggin

I though it would mean udnfined

or is that only when it divdied by 0

divides

[worthy tree]

bruh

f(-3) = (-3+3)^2 = 0^2 = 0*0 = 0

right?

[eager lynx]

yes

[worthy tree]

so u agree that the set of real numbers (-infty, infty) can be a domain of f(x)?

[eager lynx]

Yes

[worthy tree]

Ok

[eager lynx]

However for the range it's x>0 right

[worthy tree]

x>=0

[eager lynx]

okay

Is there a way to write that in interval notation

like (0, inf)

[worthy tree]

[0, infty)

[ not (

[eager lynx]

My bad

[worthy tree]

no worries

r u familiar with the notation f:A -> B

[eager lynx]

I am not

[worthy tree]

A, B are sets

A represents the domain, B the range

actually

[eager lynx]

What

So its just the reverse

[worthy tree]

B the codomain

[eager lynx]

I am unsure about what codomain means

[worthy tree]

or do u use the term image

if not thats also ok

the idea is that not all functions are onto

do u know what onto means

[eager lynx]

No I do no

not

[worthy tree]

its similar to one-to-one

for example the function f(x)=(x+3)^2 where the domain is R (which ill use to denote the real numbers) is not onto in the codomain R

because not every real number is an output of f (take the negative numbers for example)

however, if u restrict the codomain to [0, infty), then f is onto

because every element y in [0, infty) is associated with an element x in the real numbers for which f(x) = y

[eager lynx]

Is there more? if not, I am getting confused hahaha

[worthy tree]

yeah

because for a function f:A->B to have an inverse, it has to be one-to-one and onto

cuz by definition, a function g is an inverse of f if g has domain B and codomain A (g:B->A), and f(g(x)) = x for all x in B, and g(f(x)) = x for all x in A

for what u said before: $g(x)=\sqrt{x}-3$ is an inverse of $f(x)=(x+3)^2$ (in a certain domain and codomain), as we have $f(g(x))=(\sqrt{x}-3+3)^2=x$ and $g(f(x))=\sqrt{(x+3)^2}-3=x$

[eager lynx]

yes'

wait so we doing fog and gof

[solid kiln]

Bob the Builder

[worthy tree]

exactly

if f and g r inverses, then fog and gof are what we call identity functions

functions that map elements to themselves

fog has domain and codomain B, and gof A

[eager lynx]

So I have to do all that in order to get domain/range?

[worthy tree]

well what u call the range of f is the set of elements in the codomain that have an element of f that map to it

[eager lynx]

tbh Im still unsure what codomain is

[worthy tree]

yeah ig its not that important if u want to solve ur problem

[eager lynx]

I dont think I am on that yet

so I'll refrain from learning it for now

Can you show me your steps to finding out the domain/range?

cause I am just guaging it honestly and not using any math in particular

[worthy tree]

the only thing u need to do is restrict the domain to make the function one-to-one

graphically this means pass the horizontal line test

[eager lynx]

yep

[worthy tree]

so create a section of the graph of (x+3)^2 with this property

can u think of one?

[eager lynx]

Are you asking me to create a one to one function?

[worthy tree]

yeah

cuz (x+3)^2 with domain R is not one
but (x+3)^2 with some other domain could be

[eager lynx]

[-3, inf)

[worthy tree]

um

yeah

[eager lynx]

Or do you want me to say like excluding stuff below -3

[worthy tree]

that works

[eager lynx]

oh okay

[worthy tree]

so whats the range then

[eager lynx]

[0,inf)

[worthy tree]

yeah

so the inverse just swaps the domain and range

[eager lynx]

wait

let review rq

so we said f(x) is
[-3, inf)
[0, inf)

so that means

the inverse

would be
[0, inf)
[-3, inf)

In that order

[worthy tree]

yeah g: [0,inf) -> [-3, inf) defined as g(x) = sqrt(x)-3

[eager lynx]

wait thats it

Is that really it?

[worthy tree]

yeah

[eager lynx]

kay let me try this

okay

give me a little bit

of time'

[-3, inf)
[0, inf)

and then the opposite

[worthy tree]

f(-1) = 1 = f(-1/2) so its not one-to-one

[eager lynx]

is that fog

nvm

I am unsure where I went wrong

[worthy tree]

have u graphed f?

[eager lynx]

Oh I am not using a calc

I just assumed since it's +3 it goes left 3

so when x=-3, y=0

So thats not correct?

[worthy tree]

um no

theres a left translation yes

but x -> x+3 shifts it left by 3

4x -> 4x+3 is just x -> x + 3/4

[eager lynx]

correct me if I am wrong but the 4 causes a vertical stretch right

so I have to make nothing on x?

[worthy tree]

also no

u have to multiply the whole expression by 4, not just x

[eager lynx]

What

let says 1 replaces x

it would be 4+3

not the whole thing, no?

I dont understand how it become 3/4

f(x)

[worthy tree]

huh

just graph the function f

itll help

wheres the lowest point?

[eager lynx]

I initially thought it would be 0 but now 3/4

but I dont know why

[worthy tree]

huh

is the lowest y-value 3/4?

[eager lynx]

tbh I dont understand

it jsut feels like guessing now

.close

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[full dock]

how do i start with part ai)?

[knotty locust]

carefully

[ionic pendant]

do you know the formula for the area of a circular sector?

[full dock]

https://tenor.com/view/shocked-surprised-gasp-what-cat-shock-gif-13281859261409296582

[full dock]

yes

[knotty locust]

[ionic pendant]

what is it?

[full dock]

1/2 r^2 theta

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[full dock]

@whole coral chaaaaaart

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[lyric vault]

hey good morning, im learning functions on my own but i'm not getting this part, he was talking about roots in functions and suddenly he puts this on the screen. btw fur means for (it's german), second pic is the whole slide

[lyric vault]

im mainly not getting that trident shaped thing there in the curly braces

what does that mean??

[prisma stone]

It means "x is a part of Rational numbers"

x E Z would mean it is a part of integers

[lyric vault]

ok what's integers? srry english isn't my first language

[prisma stone]

Whole numbers like -1, 0, 200 etc

[lyric vault]

ohh so no 1,2?

[prisma stone]

Yes

[lyric vault]

ohh thank you makes much more sense now

[brazen needle]

Hope this helps

[lyric vault]

yea thanks!

[brazen needle]

.close

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[inland dew]

How would you get to a)

[inland dew]

i tried sin(2x + 2x) sin(3x + 1x) sin(4x+0)

[rotund grove]

what is sin 2x?

[inland dew]

ah double angle form?

[rotund grove]

sin(4x) = 2 sin2x cos 2x
= 4 sinx cosx (cos^2x-sin^2x)

[pliant junco]

2sinx.cosx

[rotund grove]

wait

option A you don't need to expand cos 2x

[pliant junco]

2sin2x.cos2x = sin4x

[rotund grove]

eyes sir!

[inland dew]

hmm

[pliant junco]

huh?!

[rotund grove]

yes*

[pliant junco]

wanna get a 4 infront boss

[rotund grove]

what's tjat

[inland dew]

i see thank you

.solve

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[vernal kiln]

hey is there anyone in grade 7?

[celest hemlock]

Why?

[vernal kiln]

imma need some help

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[thorn heart]

Just ask the question, if it's from grade 7 I assure you someone'll be able to help ya out

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[pearl lotus]

How can i show bijection between these problems
First is the total no of rooted binary trees with n nodes
Second is total no of lattice paths from 0,0 to n, n which pass below from y = x

[raw magnet]

they both trivially count the catalan numbers

just draw out the few cases

[pearl lotus]

Trivially?

[raw magnet]

for binary trees

[raw magnet]

like you can see it easily

[pearl lotus]

Yea but how can i prove that a bijection exists

[raw magnet]

if they both count the same object

show injectivity

or surjectivity

[pearl lotus]

Well for the latice paths this is the condition that at any moment no of rights are always greater than no of ups
If I can show that this condition exists for binary trees I'm done

but how

Or maybe I can use that in lattice paths the no of ups and rights are always equal and so are in binary trees but idk if this is a valid bijection

[raw magnet]

display 😭

ur thinking of this incorrectly

do you know what injective/surjective means

before we move on

[pearl lotus]

Injectivity? Like that in functions

[raw magnet]

yes

[pearl lotus]

Yea ik what abt it

[raw magnet]

show injectivity or surjectivity

and i think surjectivity is easier here

[pearl lotus]

Between the Recurrences?

[raw magnet]

no between the functions

total no of rooted binary trees with n nodes

total no of lattice paths from 0,0 to n, n which pass below from y = x

[pearl lotus]

They are the same both are catlan no
I want to know intuitively how the two problems are same

[raw magnet]

well we are trying to show that formally right now no

[pearl lotus]

Btw by reccurence will give the function

[pearl lotus]

No

intuitively

Why both prob are same?

[raw magnet]

isn't the intuition kinda obvious though

when you count proper binary trees

[pearl lotus]

I m dumb

[raw magnet]

let's say draw the case for n = 1

we define it as 1

what's the case for paths from 0, 0 to 1, 1

[pearl lotus]

1

[raw magnet]

okay now in general before we go to n,n in paths

let me draw this so it's easier

give me a second

[neon dirge]

If we want to construct a binary tree with n nodes, consider placing or not placing a node as a binary decision

You start with an empty tree

[pearl lotus]

Idt u understand my prob
Consider the lattice path prob and parenthesis prob
In parenthesis prob the no of left parenthesis are always greater than Or equal to right ones
And in the lattice prob the same condition exists so there is a bijection between the prob
Does a same condition exist in binary trees too?

[neon dirge]

Your first decision is to place or not place a node, but since the tree is currently empty and you yet have to place n nodes, you must choose Yes

Then comes the second decision, let's say in order by going the left leaf first, you can either place a node as left leaf or not

Then again, you can either place a node on the Leftmost tree position or not

If you decide not to, you move one step to the right and ask the same binary decision again

Consider the current state as position and how many nodes you have placed

You start at (0,0) because you placed 0 nodes in the tree and you are at position 0

So you must place a node

Now your position is (1,1)

The position 1 meaning we look at the left leaf of the one existing node

Position 2 would be looking at the right leaf of the existing node

So now let's say you don't place the left leaf

Then your new state is (1, 2)

There are 1 nodes in the tree and you are at position 2

So we look at the right child and have to place a node

Because we haven't yet placed n nodes

So we get to (2, 2)

Now we look at the left child of the newly placed node (position 2)

(position 3 would be right child of the newly placed node)

And we can again decide

Let's say we place a node

Then we're at (3,2)

3 nodes placed, position 2

[pearl lotus]

Hmm ok I get it

[neon dirge]

You might notice, these binary decisions mimic the lattice behavior

And here comes the crucial part!

Note we can't get to position (3, 5) for instance

[pearl lotus]

So let's consider position (x, y)
X is the no of nodes places and y is the position

[neon dirge]

Because it would mean we placed 3 nodes and are at position 5

But there can't exist 5 positions with 3 nodes in a tree

[pearl lotus]

Yup

[neon dirge]

That's why the restriction in the lattice (below y = x) is there

It prevents illegal node placement

That's why each decision in the binary tree is equivalent to one move in the lattice

reaching (n, n) would mean we placed n nodes and we're at the last position, the construction of the binary tree is done

since you can translate the problem into each other, they calculate the same Catalan number as mentioned. The translation is the bijection you seek

Since there's a unique problem for each natural n on either side, you cover all problems on both sides, the translation is injective & surjective = bijective

[pearl lotus]

yes bro I get it now
The node restricts the position
So like in the lattice paths y was always less than Or equal to x
Here we have the no of nodes placed always less than Or equal to position we are at

[neon dirge]

yop 🦇

[neon dirge]

Wait other way around

[pearl lotus]

Nice so there is a clear bijection

[neon dirge]

Position is restricted by no of nodes

[pearl lotus]

ohh wait yes mb

[neon dirge]

np

[pearl lotus]

Thanks alot
I could have never figured this out without your help

now it's much more clear

[neon dirge]

if you write it formally, for the bijection imagine being given one instance of a problem and translating it into an instance of the other

once the translation is done, one only has to show that this covers all possible instances on both sides :)

[pearl lotus]

One instance of prob?

[neon dirge]

yes "How many binary trees of n nodes exist"

Gets translated to "How many lattice paths from (0,0) to (n,n) below x=y exist"

[pearl lotus]

Yea but how could have someone figured out that position node relation

is same as y and x relation

[neon dirge]

It's a bit tricky indeed

The lattice pattern is often used for binary decisions

So I kind of had a head start having that in mind c:

[pearl lotus]

Wdym?

[neon dirge]

You can project the creation of different structures like binary trees onto lattice paths

Because in a lattice path you always just choose left or right

until you reach some goal position

[pearl lotus]

Hmm and in binary tree u choose which position to grow the tree from

And the node

[neon dirge]

Yup

[pearl lotus]

Hmm nice

[neon dirge]

The binary decision is always [Place Node] or [Move position]

[pearl lotus]

Yop

Thanks again
I take my leave now cya

.close

[trim jolt]

Channel closed

Closed by @pearl lotus

Use .reopen if this was a mistake.

[neon dirge]

np! Gl

[trim jolt]

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[wind estuary]

whats a domain and range?

[pearl lotus]

Domain is all values of input for which function gives a defined output

Range is all defined outputs

Range is a subset of Codomain

[prime lynx]

Dam I was looking for the word

"defined"

[pearl lotus]

[prime lynx]

Lol

[wind estuary]

Guyss

How do I make table of values?

for no.1

[pearl lotus]

I was reading above messages by lunatic and getting the idea in my head again and again

then I saw someone ask smthng now I'm here

[prime lynx]

Sub in the values of x within the inequality

[pearl lotus]

Haven't seen a better handwriting in my life

[prime lynx]

Into f(x)

[wind estuary]

bro that aint my handwriting

that's my teachers handwriting