Question

identifying characteristics of a function and its inverse
the following tables represent a function and its inverse. compare the functions.
which statements describe the functions? check all that apply.
□ the domain of ( h^{-1}(x) ) is the range of ( h(x) ).
□ the range of ( h^{-1}(x) ) is the range of ( h(x) ).
□ the ( x )-intercept of ( h^{-1}(x) ) is the ( x )-intercept of ( h(x) ).
□ the ( x )-coordinate of the ( x )-intercept of ( h^{-1}(x) ) is the ( y )-coordinate of the ( y )-intercept of ( h(x) ).
□ the maximum of ( h(x) ) is the largest ( x )-value of ( h^{-1}(x) ).

( h(x) )

( x )	( y )
( -2 )	( 1 )
( 0 )	( 6 )
( 2 )	( 11 )
( 4 )	( 16 )

( h^{-1}(x) )

( x )	( y )
( 1 )	( -2 )
( 6 )	( 0 )
( 11 )	( 2 )
( 16 )	( 4 )

Explanation:

To solve this, we analyze the properties of a function and its inverse:

Step 1: Recall Inverse Function Properties

For a function \( h(x) \) and its inverse \( h^{-1}(x) \):

• The domain of \( h^{-1}(x) \) is the range of \( h(x) \).
• The range of \( h^{-1}(x) \) is the domain of \( h(x) \).
• The \( x \)-intercept of \( h^{-1}(x) \) corresponds to the \( y \)-intercept of \( h(x) \) (since \( (a,0) \) on \( h^{-1}(x) \) implies \( (0,a) \) on \( h(x) \)).
• The \( x \)-coordinate of the \( x \)-intercept of \( h^{-1}(x) \) is the \( y \)-coordinate of the \( y \)-intercept of \( h(x) \).

Step 2: Analyze Each Statement

1. "The domain of \( h^{-1}(x) \) is the range of \( h(x) \)."

From inverse function properties, this is true. The domain of \( h^{-1} \) (input values) is the range of \( h \) (output values of \( h \)).

1. "The range of \( h^{-1}(x) \) is the range of \( h(x) \)."

The range of \( h^{-1} \) is the domain of \( h \), not the range of \( h \). Thus, this is false.

1. "The \( x \)-intercept of \( h^{-1}(x) \) is the \( x \)-intercept of \( h(x) \)."

The \( x \)-intercept of \( h^{-1}(x) \) (where \( y=0 \)) corresponds to the \( y \)-intercept of \( h(x) \) (where \( x=0 \)), not the \( x \)-intercept of \( h(x) \). Thus, this is false.

1. "The \( x \)-coordinate of the \( x \)-intercept of \( h^{-1}(x) \) is the \( y \)-coordinate of the \( y \)-intercept of \( h(x) \)."

For \( h(x) \), the \( y \)-intercept occurs at \( x=0 \). From the table of \( h(x) \), when \( x=0 \), \( y=6 \) (so \( (0,6) \) is the \( y \)-intercept of \( h(x) \)). For \( h^{-1}(x) \), the \( x \)-intercept occurs at \( y=0 \) (i.e., \( (a,0) \) on \( h^{-1}(x) \)), which implies \( (0,a) \) on \( h(x) \). Here, \( a=6 \), so the \( x \)-intercept of \( h^{-1}(x) \) is \( (6,0) \), and its \( x \)-coordinate (6) matches the \( y \)-coordinate of \( h(x) \)’s \( y \)-intercept (6). Thus, this is true.

1. "The maximum of \( h(x) \) is the largest \( x \)-value of \( h^{-1}(x) \)."

The maximum of \( h(x) \) (largest \( y \)-value) is the largest \( x \)-value of \( h^{-1}(x) \) (since the range of \( h \) is the domain of \( h^{-1} \)). From \( h(x) \)’s table, the largest \( y \)-value is \( 16 \) (when \( x=4 \)). From \( h^{-1}(x) \)’s table, the largest \( x \)-value is \( 16 \) (when \( y=4 \)). Thus, this is true.

Answer:

The true statements are:

• The domain of \( h^{-1}(x) \) is the range of \( h(x) \).
• The \( x \)-coordinate of the \( x \)-intercept of \( h^{-1}(x) \) is the \( y \)-coordinate of the \( y \)-intercept of \( h(x) \).
• The maximum of \( h(x) \) is the largest \( x \)-value of \( h^{-1}(x) \).

(In checkbox form, these correspond to the first, fourth, and fifth statements.)