How do you find (f\cdot g)(x) and (g\cdot f)(x) and

How do you find $\left(f\cdot g\right)\left(x\right)$ and $\left(g\cdot f\right)\left(x\right)$ and determine if the given functions are inverses of each other $f\left(x\right)={x}^{2}-3$ and $g\left(x\right)=\sqrt{x}+3$?
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Timothy Wolff
To find $f\left(g\left(x\right)\right)$ and $g\left(f\left(x\right)\right)$ substitute, g(x) for x in f(x), and f(x) for x in g(x), respectively. If they are inverses, both substitutions will equal x.
Explanation:
$f\left(g\left(x\right)\right)={\left(\sqrt{x}+3\right)}^{2}-3$
$f\left(g\left(x\right)\right)=x+6\sqrt{x}+9-3$
$f\left(g\left(x\right)\right)=x+6\sqrt{x}+6$
$g\left(f\left(x\right)\right)=\sqrt{{x}^{2}-3}+3$
They are not inverses, because both cases must reduce to x.

Ella Williams
Couldnt