https://file.aiquickdraw.com/mathbot/file/17285740908249tgfvxth.jpg

The graph shown represents a linear function. To answer the questions provided, we can proceed step by step.

### 1. Domain and Range:
- **Domain**: The domain of a function is the set of all possible input values (x-values). For this linear function, the graph extends infinitely in both directions along the x-axis, so the domain is all real numbers:
  \[
  \text{Domain}: (-\infty, \infty)
  \]

- **Range**: The range is the set of all possible output values (y-values). Similarly, the line extends infinitely in both directions along the y-axis, so the range is also all real numbers:
  \[
  \text{Range}: (-\infty, \infty)
  \]

### 2. Determining if the Function is Invertible:
A function is invertible if it is **one-to-one**, meaning that each x-value maps to exactly one unique y-value, and no horizontal line crosses the graph more than once. Since this is a linear function, and linear functions are always one-to-one, the function is invertible.

### 3. Domain and Range of the Inverse:
For an invertible function, the domain and range of its inverse are swapped:
- The **domain of the inverse** is the range of the original function.
- The **range of the inverse** is the domain of the original function.

Since both the domain and range of the original function are all real numbers, the domain and range of the inverse will also be:
\[
\text{Domain of the inverse}: (-\infty, \infty)
\]
\[
\text{Range of the inverse}: (-\infty, \infty)
\]

Would you like more details on any step or have any further questions?

Here are 5 questions to expand on this topic:
1. What is the significance of a function being one-to-one?
2. How can you algebraically determine if a function is invertible?
3. How do you find the inverse of a linear function?
4. How do the domain and range of a function compare to those of its inverse in general?
5. Can a quadratic function be invertible?

**Tip**: For linear functions, the slope gives important information about the function's behavior—whether it increases or decreases and how steeply.

Determine the domain and range of the function represented by the graph and check if it is invertible.

Learn how to determine the domain and range of a linear function from its graph. This problem also covers the concept of invertibility and explores how the domain and range of the inverse function are related to the original function. Suitable for 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