Question

a function and its inverse are shown on the same graph.
complete the sentences to compare the functions.
the y - intercept of f(x) has the same value as the
x - intercept of f inverse
y - intercept of f inverse
x - intercept of f
range including all values
, therefore, the
es also.

Explanation:

To determine the relationship between the \( y \)-intercept of \( f(x) \) and the other function values, we use the property of inverse functions: the \( y \)-intercept of a function \( f(x) \) (where \( x = 0 \)) corresponds to the point \( (0, f(0)) \) on \( f(x) \). For the inverse function \( f^{-1}(x) \), the \( x \)-intercept occurs where \( y = 0 \), i.e., \( f^{-1}(x) = 0 \) implies \( x = f(0) \) (since \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \)).

Step 1: Recall the property of inverse functions

The \( y \)-intercept of \( f(x) \) is the point \( (0, b) \) where \( f(0) = b \). For the inverse function \( f^{-1}(x) \), the \( x \)-intercept is the point \( (b, 0) \) because \( f^{-1}(b) = 0 \) (since \( f(0) = b \) implies \( f^{-1}(b) = 0 \)).

Step 2: Relate \( y \)-intercept of \( f(x) \) to \( f^{-1}(x) \)

The \( y \)-intercept of \( f(x) \) (value of \( f(0) \)) is the \( x \)-intercept of \( f^{-1}(x) \) because when we reflect the point \( (0, b) \) over the line \( y = x \) (the line of symmetry for a function and its inverse), we get \( (b, 0) \), which is the \( x \)-intercept of \( f^{-1}(x) \).

Answer:

The \( y \)-intercept of \( f(x) \) has the same value as the \( \boldsymbol{\text{x-intercept of } f^{-1}(x)} \).