Question

identify the graph of the inverse of the function g(x) shown here. g(x) = 2x² - 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 \( g(x)=2x^{2}-8 \), first, let \( y = 2x^{2}-8 \). Swap \( x \) and \( y \): \( x = 2y^{2}-8 \). Then solve for \( y \): \( 2y^{2}=x + 8 \), \( y^{2}=\frac{x + 8}{2} \), \( y=\pm\sqrt{\frac{x + 8}{2}} \). This is a square - root function (or two square - root functions, one positive and one negative), which represents a parabola that opens to the right (since it is of the form \( y^{2}=4p(x - h) \) type, here opening along the x - axis).

Step2: Analyze the original function's graph

The original function \( g(x)=2x^{2}-8 \) is a parabola opening upwards with vertex at \( (0,-8) \). When we take the inverse, the graph should be a parabola opening to the right (because we swapped \( x \) and \( y \), changing the axis of symmetry from vertical to horizontal). Looking at the options, the graph that is a parabola opening to the right (the upper right and lower right curves, which is the graph of a square - root function with both positive and negative branches) is the one that is the reflection over \( y = x \) of the original upward - opening parabola. The other option (the downward - opening parabola) is not a reflection over \( y=x \) of the original function. The graph with the two orange curves opening to the right (along the x - axis) is the graph of the inverse function.

Answer:

The graph of the inverse of \( g(x)=2x^{2}-8 \) is the graph with the two orange curves opening to the right (the one with the x - axis from - 10 to 10 and y - axis with a part showing the two branches of the square - root function).