Find two functions f(x) and g(x) such that f[g(x)] = x but g[f(x)] does not equal x

Question

anonymous · Answer

I wouldn't even know where to begin here.

anonymous · Answer

@whpalmer4

anonymous · Answer

@bahrom7893

anonymous · Answer

@inkyvoyd

anonymous · Answer

@amistre64

anonymous · Answer

@Luigi0210

zepdrix · Answer

I guess we should try to avoid using simple linear functions.
A composition of linear functions will always give us g(f(x))=x if it already gave us f(g(x))=x.

So we need something a little bit fancier.
Like a function that is `not one-to-one`.

anonymous · Answer

I tried arctan and tan, but she (my teacher) said they'd both come to x.

Wolfram alpha says she's wrong though?

zepdrix · Answer

Hmm, ya seems like she's wrong >.<
I have another idea.. how bout these two functions :)

zepdrix · Answer

$$\large\rm f(x)=x^2,\qquad\qquad\qquad g(x)=\sqrt{x}$$

anonymous · Answer

Those both equal x

zepdrix · Answer

$$\large\rm g(f(x))=\sqrt{x^2}=|x|$$

anonymous · Answer

Ah, nevermind

I just saw that

zepdrix · Answer

$$\large\rm f(g(x))=\sqrt{x}^2=x$$When the square is on the outside we have something else going on like we see here :)

zepdrix · Answer

But it's happening because x^2 is not one-to-one.
Seems like tangent should give you some similar shenanigans, but i dunno >.<

anonymous · Answer

Yeah...

Thank you! You've helped bunches1

anonymous · Answer

*!

zepdrix · Answer

cool c: