A problem related to the composition: $f^{-2}\circ g^2$

Question

I need help right here..

Question: Given: $$f(x) =3x-1\,, \, g(x) =2x +3$$ find: $$f^{-2} \circ g^2$$

Answer 1

You can always refer to the Wikipedia page. We have for a function $f(x) $, $$f^n(x) = \underbrace{f\circ f \circ \cdots f}_{n\text{ times}}$$ $$f^{-n} (x) =(f^{-1})^n$$ where $f^{-1}(x)$ is the inverse function, provided it exists.

We have: $$g^2(x) =g(g(x))=2(2x+3)+3 =4x+9$$ $$f^{-2}(x) = (f^{-1}(x))^2 = f^{-1}(f^{-1}(x)) = \frac{\frac{x+1}3+1}3 = \frac{x+4}9$$

I leave the last part to you.

Answer 2

$f(x)=3x-1$ implies $f^{-1}(x)=\frac {x+1}{3}$ and $f^{-2}(x)=\frac {x+4}{9}$

$g(x)=2x+3$implies $g^{2}(x)=4x+9$

Thus $f^{-2}og^2 =\frac {4x+13}{9}$