How to find $g^2(x)$ from $f^{-1}g(x)$ and $g(f(x))$ Given that $f^{-1}g(x)=3x-2$ and $gf(x)=12x-8$. Find $g^2(x)$.
Can anyone give me some hints on this question?
 A: HINT
$gf(f^{-1}g(x))= g(f(f^{-1}(g(x))))=g(id(g(x)))=g(g(x))=g^2(x)$
A: The high school precalculus world makes the following definitions:
$f^{-1}$ is the function that is the inverse of the one-to-one function $f$.
$fg$ (also written $f\times g$, but with a dot instead of $\times$) is defined to be the function named $(fg)$, so that function $(fg)(x)=f(x)g(x)$.
$(f\circ g)$ is defined to be the function named $(f\circ g)$, so that the function $(f\circ g)(x)$ = $f(g(x))$.
$f^n(x)=(f(x))^n, n\ne-1.$
With these standard high school calculus definitions in mind, it seems reasonable to assume that the previous Answer is flawed, making assumptions not necessarily valid in the OP's problem--i.e., maybe OP omitted some parentheses and @Bram is just trying to help. 
Parentheses make all the difference. Is it multiplication or composition where two function names are adjacent (in two places)? Is the 2 an exponent or does it imply $g\circ g$?
I assume the problem wants an algebraic answer in terms of $x$ alone. I think any posted Answer should be preceded by having acquired such an answer.
I've been working on this for long enough to know that I need clarification. There are at least four possible problems to work, with and without parentheses where legal or omitted as in given problem statement.
BTW, as written and interpreting it in terms of the above definitions, the problem becomes:
Given $f^{-1}(x)g(x)=3x-2$ and $g(x)f(x)=12x-8$. Find $(g(x))^2$.
I sincerely doubt that's the precise problem statement or is equivalent to it. Surely the first equation should begin $f^{-1}(g(x))$.
