Question

is the inverse of g(x) a function? use the drop - down menus to explain.

(g(x)=x^{2}-2)

click the arrows to choose an answer from each menu.

the graph of the inverse of (g(x)) is the reflection of the graph of (g(x)) across the choose... . the inverse of (g(x)) choose... a function because for each input of the inverse of (g(x)) there is choose... one unique output.

Explanation:

Step1: Identify reflection line

The inverse of a function's graph is reflected across the line $y=x$.

Step2: Test if inverse is a function

First, find the inverse of $g(x)=x^2-2$:
Set $y=x^2-2$, swap $x$ and $y$: $x=y^2-2$, solve for $y$: $y=\pm\sqrt{x+2}$.
For a single input $x> -2$, there are two outputs, so it fails the vertical line test.

Step3: Determine output uniqueness

Since one input maps to two outputs, there is NOT one unique output.

Answer:

The graph of the inverse of $g(x)$ is the reflection of the graph of $g(x)$ across the $\boldsymbol{y=x}$ line. The inverse of $g(x)$ $\boldsymbol{is not}$ a function because for each input of the inverse of $g(x)$ there is $\boldsymbol{not}$ one unique output.