is the inverse of g(x) a function? use the drop - down menus to explain.\n\n(g(x)=x^{2}-2)\n\nclick the…

is the inverse of g(x) a function? use the drop - down menus to explain.\n\n(g(x)=x^{2}-2)\n\nclick the arrows to choose an answer from each menu.\n\nthe 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.

is the inverse of g(x) a function? use the drop - down menus to explain.\n\n(g(x)=x^{2}-2)\n\nclick the arrows to choose an answer from each menu.\n\nthe 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.

Answer

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).