If $f(x) = x-5$ and $g(x) = x^2 -5$, what is $u(x)$ if $(u \circ f)(x) = g(x)$?

Question

Let $f(x) = x-5$, $g(x) = x^2 -5$. Find $u(x)$ if $(u \circ f)(x) = g(x)$.

I know how to do it we have $(f \circ u) (x)$, but only because $f(x)$ was defined. But here $u(x)$ is not defined. Is there any way I can reverse it to get $u(x)$ alone?

Answer 1

Hint: We are told that $u(x-5)=x^2-5$. Let $t=x-5$. Now express $x^2-5$ in terms of $t$.

Answer 2

Hint: $$x^2-5=(x-5)(x+5)$$ $$(x+5)=(x-5)+10$$

Answer 3

I think I figured out what my professor did now . . .

$(u \circ f)(x) = g(x)$

$(u \circ f)(f^{-1} (x)) = g( f^{-1}(x)) $

$\big((u \circ f) \circ f^{-1}\big)(x) = (g \circ f^{-1})(x) $

$\big(u \circ (f \circ f^{-1})\big)(x) = (g \circ f^{-1})(x) $

$u(x) = g(f^{-1}(x))$

$u(x) = g(x+5)$

I think this is right. Please correct me if I'm wrong.