https://file.aiquickdraw.com/mathbot/file/1727980477492c8up12a0.jpg

The problem asks to restrict the domain of the given function so that it becomes one-to-one, which will allow the function to have an inverse. 

Looking at the graph, it is a V-shaped function that increases, reaches a peak, and then decreases. To make the function one-to-one, we need to select a portion of the graph where the function is either strictly increasing or strictly decreasing, since a function must pass the Horizontal Line Test (no horizontal line should intersect the graph more than once) to have an inverse.

### Observing the graph:
- The function reaches its peak at \( x = 0 \) and \( y = 3 \).
- To make the function one-to-one, we can restrict the domain to either:
  - The left side of the peak (where the function is decreasing).
  - The right side of the peak (where the function is increasing).

### Potential domain restrictions in interval notation:
1. **For the increasing portion**: The domain would be \( [0, 3] \), as the function is strictly increasing from \( x = 0 \) onward.
2. **For the decreasing portion**: The domain would be \( [-3, 0] \), as the function is strictly decreasing in this range.

### Correct answers:
The valid domain restrictions that make the function one-to-one are:
- \( [-3, 0] \)
- \( [0, 3] \)

Would you like further explanation on how domain restrictions work or have any other questions? Here are some related questions to explore this further:

1. What is the Horizontal Line Test and how does it determine if a function is one-to-one?
2. Why can't a function with a V-shape have an inverse without domain restriction?
3. How does restricting the domain affect the range of the inverse function?
4. Can a function have more than one valid domain restriction for it to have an inverse?
5. What is the significance of interval notation when discussing domains and ranges?

**Tip:** When determining if a function has an inverse, always check if it passes the Horizontal Line Test—this ensures that the function is one-to-one, a key requirement for an inverse to exist.

You want to restrict the domain of the function shown in the graph below to make it one-to-one so that it will have an inverse. What domain, in interval notation, could you use?

Learn how to restrict the domain of a function to make it one-to-one, allowing it to have an inverse. This problem covers key concepts in inverse functions, domain restriction, and the Horizontal Line Test. Suitable for high school students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation