Calculating the inverse of the given function:$g(x)=f(x)-2$ in terms of $f^{-1}$

Question

As the title states I have to find the inverse in terms of $f^{-1}$ for $g(x)=f(x)-2$

I'm having a little hard time understanding how treat this function.

I've read few examples here on the math.stackexchange and there it's solved by treating it as a composite.

However, the author of this book did abstain from using composites which leads me believe that it's possible without, even though it might be more difficult.

Anyhow, in the formula the author gets the answer:

$f(y)=x+2$ and $g^{−1}(x) = y = f^{−1}(x + 2)$

I would very much appreciate it if somebody could provide some insight.

Answer 1

Guide:

Try to solve

$$y = f(x)-2,$$

that is express $x$ in terms of $y$.

Answer 2

$$g(x) = f(x) - 2$$ Thus, subbing in $g^{-1}(x)$ for $x$, $$g(g^{-1}(x)) = x = f(g^{-1}(x)) - 2 \implies f(g^{-1}(x)) = x + 2$$ so, composing this with $f^{-1}$, $$f^{-1}(f(g^{-1}(x))) = g^{-1}(x) = f^{-1}(x+2)$$