For the following composite functions, find possible formulas for $g(x)$, $h(x)$, and $f(x)$, as indicated below. Assume $g(x) \neq x$, $h(x) \neq x$, and $f(x) \neq x$.

a) Find a formula for $g(x)$ given that $g(h(x))=(x+6)^2$ and $h(x)=x+6$.
$g(x)=$

b) Suppose $g(h(x))=\sqrt{6x^2+6}$.
i) Find a formula for $g(x)$ if $h(x)=6x^2$
$g(x)=$

ii) Find a formula for $g(x)$ if $h(x)=x^2$.
$g(x)=$

iii) Find a formula for $h(x)$ if $g(x)=\sqrt{x}$.
$h(x)=$

c) Find a formula for $h(x)$ given that $g(h(x))=\dfrac{1}{x-1}+x^2-2x+1$ and $g(x)=\frac{1}{x}+x^2$.
$h(x)=$

d) Suppose $f(g(h(x)))=\dfrac{1}{\log{(x^2+2)}}$.
i) Find a formula for $f(x)$ if $g(x)=\log(x+2)$ and $h(x)=x^2$.
$f(x)=$

ii) Find a formula for $g(x)$ if $f(x)=\frac{1}{\log(x)}$ and $h(x)=x^2$.
$g(x)=$

iii) Find a formula for $h(x)$ if $f(x)=\frac{1}{x}$ and $g(x)=\log(x)$.
$h(x)=$

e) Suppose $g(h(x))=\left(\ln(x)\right)^{2/3}$.
i) Find a formula for $g(x)$ if $h(x)=\ln(x)$.
$g(x)=$

ii) Find a formula for $h(x)$ if $g(x)=x^2$.
$h(x)=$

f) Find a formula for $g(x)$ given that $g(h(x))=\dfrac{1}{(x-3)^2}$ and $h(x)=(x-3)^2$.
$g(x)=$

Hint: