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 $y=x$.

Step2: Test inverse for function property

First, find the inverse:
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 2 outputs, so it fails the vertical line test.

Step3: Fill in the logic

Since there is not one unique output for each input, the inverse is not a function.

Answer:

The graph of the inverse of $g(x)$ is the reflection of the graph of $g(x)$ across the $\boldsymbol{y=x}$. 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.