Question

given the function $f(x)=\frac{1}{2}\sqrt{x}$, use f(g(x)) to verify or disprove the proposed inverse function $g(x)=4x^2$.
the function g(x) select the inverse of f(x) because f(g(x)) select is is not

Explanation:

Step1: Substitute $g(x)$ into $f(x)$

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

Step2: Simplify the square root term

$\sqrt{4x^2} = 2|x|$, so $f(g(x)) = \frac{1}{2} \cdot 2|x| = |x|$

Step3: Compare to identity function

For $g(x)$ to be the inverse, $f(g(x))$ must equal $x$ (not $|x|$), which only holds for $x \geq 0$, not all valid $x$ for $f(x)$.

Answer:

The function $g(x)$ is not the inverse of $f(x)$ because $f(g(x))$ is $|x|$ (not equal to $x$ for all $x$ in the domain of $f(x)$).