Question

is g(x) the inverse of f(x)?

Explanation:

Step1: Recall inverse graph property

The graph of a function's inverse is the reflection of the original function's graph over the line $y=x$.

Step2: Compare graphs to $y=x$

Visually, $g(x)$ is a downward-opening parabola, while $f(x)$ is an upward-opening parabola. $g(x)$ is not the reflection of $f(x)$ across the line $y=x$; instead, it appears to be a vertical reflection (over the x-axis) rather than the reflection required for an inverse function.

Step3: Verify with function definition

Additionally, $f(x)$ fails the horizontal line test (it is a parabola, so multiple x-values map to the same y-value), meaning $f(x)$ does not have an inverse function at all, as it is not one-to-one.

Answer:

No, $g(x)$ is not the inverse of $f(x)$.