How do you find (f*g)(x) and (g*f)(x) and determine if the given functions are inverses of each other f(x) = x^2 − 3 and g(x) = sqrtx+3?

Jan 24, 2017

To find $f \left(g \left(x\right)\right) \mathmr{and} g \left(f \left(x\right)\right)$, 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.