Three functions are graphed on the coordinate plane.The x-axis goes from negative 6 to 6 and the y-axis goes from negative 6 to 6.

Question

A function labeled f left-parenthesis x right-parenthesis has points left-parenthesis 0 comma 1 right-parenthesis, left-parenthesis 1 comma 3 right-parenthesis, and left-parenthesis 3 comma 5 right-parenthesis. Starting from the point left-parenthesis 0 comma 1 right-parenthesis the graph curves right and up and then up and right.

A second function labeled g left-parenthesis x right-parenthesis has points left-parenthesis 0 comma 1 right-parenthesis, left-parenthesis 1 comma 3 right-parenthesis, and left-parenthesis 3 comma 5 right-parenthesis. Starting from the point left-parenthesis 0 comma 1 right-parenthesis the graph curves up and right and then right and up.

A third function labeled h left-parenthesis x right-parenthesis has points left-parenthesis 1 comma 0 right-parenthesis, left-parenthesis 3 comma 1 right-parenthesis, and left-parenthesis 5 comma 3 right-parenthesis. Starting from the point left-parenthesis 1 comma 0 right-parenthesis the graph curves up and right and then right and up.

Question
Use the graph to answer the question.
Which function, g or h, is the inverse of function f, and why?
(1 point)
Responses

The function h is the inverse because h and f intersect at one, and only one, point.
The function h is the inverse because h and f intersect at one, and only one, point.

The function h is the inverse because for every point left-parenthesis x comma y right-parenthesis on the graph of f, there is a corresponding point left-parenthesis y comma x right-parenthesis on the graph of h.
The function h is the inverse because for every point Image with alt text: left-parenthesis x comma y right-parenthesis on the graph of f , there is a corresponding point Image with alt text: left-parenthesis y comma x right-parenthesis on the graph of h .

The function g is the inverse because for every point left parenthesis a comma b right parenthesis on the graph of f, there is a corresponding point left parenthesis b comma a right parenthesis on the graph of g.
The function g is the inverse because for every point Image with alt text: left parenthesis a comma b right parenthesis on the graph of f , there is a corresponding point Image with alt text: left parenthesis b comma a right parenthesis on the graph of g .

The function g is in the inverse because g and f intersect at multiple points.

Answer 1

To determine which function, \( g \) or \( h \), is the inverse of function \( f \), we need to identify if for every point \( (x, y) \) on the graph of \( f \), there is a corresponding point \( (y, x) \) on the graph of \( g \) or \( h \).

Given the points on each function:

• Function \( f \) has points \( (0, 1) \), \( (1, 3) \), and \( (3, 5) \).
• Function \( g \) has points \( (0, 1) \), \( (1, 3) \), and \( (3, 5) \).
• Function \( h \) has points \( (1, 0) \), \( (3, 1) \), and \( (5, 3) \).

Let's check for inverse relationships:

1. For Function \( g \):

   • Point \( (0, 1) \): Corresponding point in \( g \) is \( (1, 0) \) → Not a match.
   • Point \( (1, 3) \): Corresponding point in \( g \) is \( (3, 1) \) → Not a match.
   • Point \( (3, 5) \): Corresponding point in \( g \) is \( (5, 3) \) → Not a match.

   Overall, \( g \) does not correspond to the points of \( f \) in an inverse relationship.

2. For Function \( h \):

   • Point \( (0, 1) \): Corresponding point in \( h \) is \( (1, 0) \).
   • Point \( (1, 3) \): This point does not appear in \( h \).
   • Point \( (3, 5) \): This point does not appear in \( h \).

   It appears that \( h \) does not cover all the points from \( f \) either.

After assessing \( g \) and \( h \), the actual matching pairs are seen when checking established inverses, confirming \( g \) does have inverse points \( (3, 1) \) and \( (5, 3) \).

Thus, the correct answer is:

The function g is the inverse because for every point \( (a, b) \) on the graph of \( f \), there is a corresponding point \( (b, a) \) on the graph of g.