Question

Let f(x)= x/x-5. Find a function y=g(x) so that (f○g)(x)=2x.

Answer 1

If I'm not mistaken, (f○g)(x) is defined as f(g(x)). So:
\[f(g(x)) = \frac{ g(x) }{ g(x)-5 } = 2x\]
Then, there must be a function g(x) such that:
\[g(x) = 2x(g(x)-5)\]

Answer 2

It's called the composition of f and g, i've never looked much into functional analysis yet, but let me try to find a g(x) :)

Answer 3

I can work through composition problems when both functions are given, I just don't know how to work through it backwards...I cannot find a value of g(x) to make f(g(x))=2x

Answer 4

Having some trouble too :p

Answer 5

I'm glad I'm not the only one

Answer 6

solving ahead,
\(g(x) = 2x(g(x)-5) \\ g(x)[1-2x]= -10x \\ g(x) = \dfrac{-10x}{1-2x}= \dfrac{10x}{2x-1} \)

Answer 7

Thank you so much! can you explain how you got that?

Answer 8

which step ?, i guess you didn't get 2nd step ?

Answer 9

Yes i arrived at the same :) It's basically taking the 2xg(x) part to the left side to bundle both g(x) terms.

Answer 10

\(g(x) = 2x(g(x)-5) \\ g(x)=2x g(x)-10x \\ g(x)-2xg(x) =-10x\\ g(x)[1-2x]= -10x \\ g(x) = \dfrac{-10x}{1-2x}= \dfrac{10x}{2x-1}\)
now ?

Answer 11

Thank you so much! I understand now, just solved another on my own!

Answer 12

i am glad to hear that :)