A function f(x)f(x) includes the points (2, 3)(2, 3), (−5, 1)(−5, 1), and (10, −3)(10, −3) in its graph. Based on this, which of the following points must be included in the graph of f−1(x)f−1(x)?

To determine which points must be included in the graph of \( f^{-1}(x) \) (the inverse of the function \( f(x) \)), we use the fact that the inverse function swaps the \( x \)- and \( y \)-coordinates of each point on the graph of \( f(x) \). 

Given the points on \( f(x) \):
- \( (2, 3) \)
- \( (-5, 1) \)
- \( (10, -3) \)

The points on the graph of \( f^{-1}(x) \) will be:
- \( (3, 2) \) (from \( (2, 3) \))
- \( (1, -5) \) (from \( (-5, 1) \))
- \( (-3, 10) \) (from \( (10, -3) \))

Thus, the points that must be included in the graph of \( f^{-1}(x) \) are:
- \( (3, 2) \)
- \( (1, -5) \)
- \( (-3, 10) \)

Would you like more details or have any questions? Here are five related questions you can explore:
1. How do you find the inverse of a function algebraically?
2. What are the conditions for a function to have an inverse?
3. How does the graph of \( f^{-1}(x) \) compare to the graph of \( f(x) \)?
4. Can every function have an inverse? Why or why not?
5. How do you verify if two functions are inverses of each other?

**Tip:** When graphing inverse functions, the line \( y = x \) serves as a mirror line for the reflection of points between the functions.

A function f(x) includes the points (2, 3), (−5, 1), and (10, −3) in its graph. Based on this, which of the following points must be included in the graph of f−1(x)?

Learn how to determine the points on the graph of the inverse of a function. Given the points (2, 3), (−5, 1), and (10, −3) on the graph of f(x), discover how to find the corresponding points on the graph of f−1(x). This problem covers inverse functions and coordinate geometry concepts, suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation