[Math] find g(x) given f(x) and the composition (g o f)(x)

Question

I've been stuck on this final math problem for ages

I'm given $$f(x) = x^2 + 1$$

and the final composition is $$(g \circ f)(x) = \frac{1}{x^2 + 4}$$

I'm asked to find that $g(x)$ was in order to make this true, but i'm not sure how?

Answer 1

Here's my reasoning:

$$ g(f(x)) = \frac{1}{x^2 + 4} = \frac{1}{(x^2+1)+3}$$

Since $x^2 + 1 = f(x)$

$$g(f(x)) = \frac{1}{f(x)+3} \implies g(x) = \frac{1}{x+3} $$

With $g(x)$, Note: $x \ne -3$ $(x \in \mathbb{R})$