Find a function $f:\bR\to\bR$ such that $\int f(f(x)+x)dx=f(f(x))+C$

[fallen gazelle]

haha.

@steel violet @solar fjord

[steel violet]

hmmmn

[solar fjord]

Ok so i told you i was gonna solve that equation

[steel violet]

f'(f(x)).f'(x)=f(f(x)+x)

[solar fjord]

Wait let me send a photo

[fallen gazelle]

wait no latex?

[solar fjord]

What?

[fallen gazelle]

Find a function $f:\bR\to\bR$ such that $$\int f(f(x)+x)dx=f(f(x))+C$$

[steel violet]

people like me dony know latex

[rich harness]

dark matter

[steel violet]

how do u typr latex so fast

[fallen gazelle]

there i thought it didnt work

[fallen gazelle]

oh i just copy pasted from the title lel

[steel violet]

oh

[fallen gazelle]

title takes maybe 10 seconds to create

[solar fjord]

Done

[fallen gazelle]

you didnt find the funny solution

[steel violet]

wait leme see

[solar fjord]

Wdym funny solution?

Hold….

[fallen gazelle]

there are more solutions than that

[solar fjord]

I know

I wrote some of them

I wrote f(x) = x

[steel violet]

f(x)=-x

?!

[solar fjord]

And f(x) = c*x ohhhhh i see it now

[fallen gazelle]

i will tell in dm

[solar fjord]

I think this was the funny solution

[fallen gazelle]

nope

[solar fjord]

Oh

Then what was it?

Anyway let’s move on to geometry

What should we do for that?

Circle?

Triangle?

Quadrilateral?

[steel violet]

i choose straight line

[solar fjord]

Ummm

Ok

If we have a straight line labeled
AB

C is in the middle of it

[steel violet]

uh lines not line sement

lines segements come under points.

whole lines in 2d

[solar fjord]

I was thinking shapes

[steel violet]

a line is technically a circle

[solar fjord]

……..

[steel violet]

its equidistant from infinity from all points

[solar fjord]

Ok let’s do algebra then

So what should we do here?

Ok ill choose

We’ll do real numbers

Starting off easy

$\sqrt{50} + 2\sqrt{2}$

[rich harness]

TabMineCrafter

[solar fjord]

@steel violet @fallen gazelle?

[steel violet]

what

[steel violet]

7root(2)

[fallen gazelle]

what

[solar fjord]

f(x) = 0

Am i right?

[fallen gazelle]

who knows

[solar fjord]

Should be i derived it

assumed f(x) = a + xb

b could only be -1 or 0

[solar fjord]

do you need?

[solar fjord]

Yep

@steel violet @fallen gazelle let’s get to trigonometry

[fallen gazelle]

no

[solar fjord]

……

Circles?

@fallen gazelle you told me to
Give a problem

On trigonometry

We have a right triangle labeled ABC, AB is 40 and AC is 30, find the sin of the angle ABC

[fallen gazelle]

ok

3/4 or sqrt(7)/4

ez guess

[solar fjord]

No

Ill draw it to make it easier

[fallen gazelle]

ok

[solar fjord]

I know my writing is messy

@fallen gazelle i drew it

[fallen gazelle]

oh ok

so sin(ABC)=3/5

[solar fjord]

Yes

We have a normal triangle labeled ABC with AB = 12, BC = 13 and AC =15, find the area of that triangle

Ok AC is 15

[toxic olive]

@fallen gazelle f(x) = Ae^x - x - 1

and f(x) = 0

[fallen gazelle]

its not the irght answer

[fallen gazelle]

yes

[toxic olive]

did you check it?

maybe a +x not a -x

[fallen gazelle]

u solved it wrong way

[toxic olive]

wat

[fallen gazelle]

show work again

[solar fjord]

What are you even solving here?

Oh nvm i found it

[toxic olive]

wa

[fallen gazelle]

show work bro

[toxic olive]

I fold

[fallen gazelle]

...

[steel violet]

https://youtu.be/n_E3bLYuQBo?si=jXuQD4mTdogAylGz

[solar fjord]

f(x) = 0 i already said

f(x) = a + bx
If you literally put everything in the identity you get
b = 0 or -1
f(x) = a or f(x) = a - x
assume a = 0 because a - x won't work at all
f(x) = 0

The initial derivation is long

Let’s do circle problems

[wraith edge]

f(x) = 0 right

[steel violet]

that's is one of the solutions

theres one more

[wraith edge]

ok but it says find a solution

therefore qed

[steel violet]

well fair

[solar fjord]

Should we do another problem?@here

[uncut vine]

tf is this notation

$\int f(f(x)+x)\mathrm{d}x=f(f(x))+C$

[rich harness]

fr

[uncut vine]

that doesnt eliminate the constant of integration

u can write something like "the antiderivative of"

or something like "d/dx f(f(x))= stuff"

e^x

LMAO e^x WORKS

NAHAHHHHH

HAHAHAHAHAH

it's the only exponential that works

[fallen gazelle]

finally you found the cheese solution

i was randomly thinking about how to solve $\int e^{e^x}dx$ then stumbled on that and realized that it was cool

[rich harness]

dark matter

[copper shadow]

Wow

[uncut vine]

fym finally, itwas my first guess

[fallen gazelle]

let me phrase it better: finally, someone found the cheese solution

[sinful rain]

0 is much more cheese than e^x

[fallen gazelle]

no

e^x is mor chez

[sinful rain]

not really