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: Recall inverse function reflection

The graph of an inverse function of a function \( y = f(x) \) is the reflection of the graph of \( f(x) \) across the line \( y = x \). So the first drop - down (for the line of reflection) should be "line \( y = x \)".

Step2: Determine if \( g(x)=x^{2}-2 \) is one - to - one

A function has an inverse that is also a function if and only if the original function is one - to - one (passes the horizontal line test). The function \( g(x)=x^{2}-2 \) is a parabola opening upwards with vertex at \( (0, - 2) \). It does not pass the horizontal line test (for example, \( g(2)=2^{2}-2 = 2 \) and \( g(-2)=(-2)^{2}-2 = 2 \), so two different inputs give the same output). So the inverse of \( g(x) \) will not be a function. To check if a relation is a function, we use the vertical line test. For the inverse of \( g(x) \), if we consider the relation, when we reflect \( g(x) \) over \( y = x \), the inverse relation will not pass the vertical line test (because the original function did not pass the horizontal line test). So the second drop - down (Is the inverse a function?) should be "is not", and the third drop - down (reason related to vertical line test) should be "does not pass the vertical line test" (because for a relation to be a function, every vertical line must intersect the graph at most once).

Answer:

1. The graph of the inverse of \( g(x) \) is the reflection of the graph of \( g(x) \) across the \(\boldsymbol{\text{line } y = x}\).
2. The inverse of \( g(x) \) \(\boldsymbol{\text{is not}}\) a function because for each input of the inverse of \( g(x) \) there \(\boldsymbol{\text{does not pass the vertical line test}}\) (i.e., there are inputs with more than one output).