Let $(f \circ g)(x) = x^2 +2x -1$. Find $f(x)$ if $g(x) = x+2$

Question

Problem : Let $(f \circ g)(x) = x^2 +2x -1$. Find $f(x)$ if $g(x) = x+2$

My Attempted Solution

$$f(g(x)) = x^2 +2x-1$$ $$f(x+2) = x^2 +2x -1$$

But that is as far as I got. The problem I'm having is that I can't seem to find a way to algebraically relate $g(x)$ to $f(g(x))$ and solve for the unknown $f(x)$.

As a generalized extension to the above problem, furthermore if I have a composition of two aribtrary functions $f,g : \mathbb{R} \to \mathbb{R}$ with $g$, known and $f$ being an unknown function, what is the general algebraic method to solve for $f(x)$ if $f(g(x))$ is given. i.e.

$$\text{If}\ \ f(g(x)) = \alpha \ \ \text{and} \ \ g(x) = \zeta \ \ $$ $$\text{How do you algebraically solve for } f(x)$$

Answer 1

Hint: $x^2 + 2x - 1 = (x+2)^2 - 2(x+2) - 1$

Answer 2

Replace $x \rightarrow x -2$ in $f(x+2) = x^2 +2x -1$