Question

select the correct answer.
consider function f.
$f(x) = \sqrt3{x - 7}$
select $g(x)$ so that $f(g(x)) = g(f(x)) = x$ for all values of $x$.
$g(x)=x^3 - 7$
$g(x)=x^3 + 7$
$g(x)=(x + 7)^3$
$g(x)=(x - 7)^3$

Explanation:

Step1: Identify inverse function goal

We need to find $g(x)$, the inverse of $f(x)=\sqrt[3]{x-7}$, such that $f(g(x))=g(f(x))=x$.

Step2: Set $y = f(x)$ and swap $x,y$

Let $y = \sqrt[3]{x-7}$. Swap variables: $x = \sqrt[3]{y-7}$.

Step3: Solve for $y$ to find inverse

Cube both sides: $x^3 = y - 7$.
Rearrange to isolate $y$: $y = x^3 + 7$.
This $y$ is the inverse function $g(x)$.

Step4: Verify composition (optional check)

Check $g(f(x)) = (\sqrt[3]{x-7})^3 + 7 = (x-7) + 7 = x$, which satisfies the requirement.

Answer:

$g(x)=x^3 + 7$