Select the correct answer.
Which statement is true about a one-to-one function?

 
The domain and range of the function is the same as the domain and range of the inverse function.
 
If the x- and y-values in each ordered pair of a function are interchanged, the resulting set of ordered pairs is known as the inverse of a function.
 
Each input value (x) corresponds to only one output value (y), and each output value corresponds to more than one input value.
 
The graph of the function passes the vertical line test, but does not pass the horizontal line test.

The correct answer is:

**The domain and range of the function is the same as the domain and range of the inverse function.**

### Explanation:
A one-to-one function is a function where each input value corresponds to exactly one output value, and each output value corresponds to exactly one input value. This property ensures that the function has an inverse, and for the inverse function, the domain and range are swapped. Therefore, the domain and range of the original function become the range and domain of the inverse function, respectively.

Other statements:
- **"If the x- and y-values in each ordered pair of a function are interchanged, the resulting set of ordered pairs is known as the inverse of a function."**  
  - This statement is also true in a general sense but it's not specific to one-to-one functions, as not every function has an inverse. Only one-to-one functions have inverses.
  
- **"Each input value (x) corresponds to only one output value (y), and each output value corresponds to more than one input value."**  
  - This is false for a one-to-one function, because each output value corresponds to only one input value in a one-to-one function.

- **"The graph of the function passes the vertical line test, but does not pass the horizontal line test."**  
  - This is false for a one-to-one function. A one-to-one function must pass both the vertical line test (to be a function) and the horizontal line test (to ensure it is one-to-one).

Would you like more details or have any questions?

---

Here are 5 related questions to expand your understanding:
1. What is the horizontal line test and how is it used to determine if a function is one-to-one?
2. How do you find the inverse of a one-to-one function?
3. Can a function be one-to-one and not have an inverse? Why or why not?
4. What are the key differences between a one-to-one function and a many-to-one function?
5. How do domain restrictions affect whether a function is one-to-one?

**Tip:** A one-to-one function guarantees that its inverse is also a function. This is why the horizontal line test is essential in identifying these functions.

Select the correct answer. Which statement is true about a one-to-one function? The domain and range of the function is the same as the domain and range of the inverse function. If the x- and y-values in each ordered pair of a function are interchanged, the resulting set of ordered pairs is known as the inverse of a function. Each input value (x) corresponds to only one output value (y), and each output value corresponds to more than one input value. The graph of the function passes the vertical line test, but does not pass the horizontal line test.

Discover how one-to-one functions work and why their domain and range match the inverse function. Learn about the vertical and horizontal line tests, and explore the inverse function theorem. 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