QuestionWhich statement describes how the graph of a function, $h(x)$ , and its inverse, $h^{-1}(x)$ , are related? The line $y=-x$ is the perpendicular bisector of each segment connecting a point on $h(x)$ to the corresponding point on $h^{-1}(x)$ . The line $y=x$ is the perpendicular bisector of each segment connecting a point on $h(x)$ to the corresponding point on $h^{-1}(x)$ . The graph of the inverse of $h(x)$ is a reflection over the line $y=0$ of the graph of $h(x)$ . The $y$ -axis is the perpendicular bisector of each segment connecting a point on $h(x)$ to the corresponding point on $h^{-1}(x)$ .

Studdy Solution

STEP 1

1. We are given a function $h(x)$ and its inverse $h^{-1}(x)$ .
2. We need to determine the relationship between the graph of $h(x)$ and $h^{-1}(x)$ .
3. We have four statements to evaluate.

STEP 2

1. Understand the properties of a function and its inverse.
2. Evaluate each statement based on these properties.
3. Identify the correct statement.

STEP 3

Understand the properties of a function and its inverse.
- The graph of a function $h(x)$ and its inverse $h^{-1}(x)$ are reflections of each other over the line $y = x$ .

STEP 4

Evaluate each statement based on the properties of a function and its inverse.
- Statement 1: The line $y = -x$ is the perpendicular bisector of each segment connecting a point on $h(x)$ to the corresponding point on $h^{-1}(x)$ . - This is incorrect because the line of reflection for a function and its inverse is $y = x$ , not $y = -x$ .
- Statement 2: The line $y = x$ is the perpendicular bisector of each segment connecting a point on $h(x)$ to the corresponding point on $h^{-1}(x)$ . - This is correct because the graph of a function and its inverse are reflections over the line $y = x$ .
- Statement 3: The graph of the inverse of $h(x)$ is a reflection over the line $y = 0$ of the graph of $h(x)$ . - This is incorrect because the reflection occurs over the line $y = x$ , not $y = 0$ .
- Statement 4: The $y$ -axis is the perpendicular bisector of each segment connecting a point on $h(x)$ to the corresponding point on $h^{-1}(x)$ . - This is incorrect because the reflection occurs over the line $y = x$ , not the $y$ -axis.

STEP 5

Identify the correct statement.
The correct statement is: "The line $y = x$ is the perpendicular bisector of each segment connecting a point on $h(x)$ to the corresponding point on $h^{-1}(x)$ ."

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