Question

identify the graph of the inverse of the function g(x) shown here.
$g(x) = 2x^2 - 8$

Explanation:

Step1: Recall inverse function graph rule

The graph of an inverse function \( g^{-1}(x) \) is the reflection of the graph of \( g(x) \) over the line \( y = x \). Also, for a function \( y = g(x) \), to find the inverse, we swap \( x \) and \( y \) and solve for \( y \).

Given \( g(x)=2x^{2}-8 \), let \( y = 2x^{2}-8 \). To find the inverse, swap \( x \) and \( y \):
\( x = 2y^{2}-8 \)

Step2: Analyze the original function's graph

The original function \( g(x)=2x^{2}-8 \) is a parabola opening upwards with vertex at \( (0, - 8) \) (wait, in the given graph, the vertex seems to be at \( (0, - 8) \) but the graph in the image has a vertex near \( y=-8 \)? Wait, the user's graph shows \( g(x) = 2x^{2}-8 \) with vertex at \( (0, - 8) \)? Wait, maybe the graph in the image has a typo, but the function is \( y = 2x^{2}-8 \), a parabola opening up.

When we find the inverse, we have \( x = 2y^{2}-8 \), which can be rewritten as \( 2y^{2}=x + 8\) or \( y^{2}=\frac{x + 8}{2}\), which is a parabola opening to the right (since it's solved for \( y^{2}\) in terms of \( x \)).

Now, looking at the options:

• The top - right graph is a parabola opening to the right (since it's symmetric about the \( x \)-axis? Wait, no, symmetric about the \( y \)-axis? Wait, no, the top - right graph has two branches, one above and one below the \( x \)-axis, symmetric about the \( y \)-axis, and opening to the right (since as \( x \) increases, \( y \) goes up and down). The bottom - right is a downward - opening parabola (opening down), and the original function is upward - opening.

The inverse of a parabola opening up (which is not one - to - one over its entire domain, but if we consider the inverse relation) will be a parabola opening to the right (since we swap \( x \) and \( y \)). The top - right graph is a parabola opening to the right, which matches the reflection over \( y=x \) of the original upward - opening parabola.

The original function \( y = 2x^{2}-8 \) (parabola opening up) has an inverse relation (not a function over the entire domain, but the graph of the inverse relation) which is a parabola opening to the right, which is the top - right graph.

Answer:

The graph of the inverse of \( g(x)=2x^{2}-8 \) is the top - right graph (the one with the two red branches symmetric about the \( y \)-axis and opening to the right, with \( x \) ranging from - 10 to 10 and \( y \) around the \( x \)-axis).