https://r2.erweima.ai/mathbot/image/17237582493234d5kendc.jpg

Evaluate the function as indicated. Determine its domain and range

The problem presents a piecewise function \( f(x) \) defined as follows:

\[
f(x) = \begin{cases} 
x + 1, & \text{if } x < 1 \\
-x + 1, & \text{if } x \geq 1 
\end{cases}
\]

You are asked to evaluate the function at specific values: \( f(-3) \), \( f(1) \), \( f(3) \), and \( f(6 + 1) \). Additionally, the domain and range of the function need to be determined.

### Evaluations

1. **\( f(-3) \)**:
   - Since \( -3 < 1 \), we use the first piece of the function: \( f(x) = x + 1 \).
   - Substituting \( x = -3 \):
   \[
   f(-3) = -3 + 1 = -2
   \]

2. **\( f(1) \)**:
   - Since \( 1 \geq 1 \), we use the second piece of the function: \( f(x) = -x + 1 \).
   - Substituting \( x = 1 \):
   \[
   f(1) = -1 + 1 = 0
   \]

3. **\( f(3) \)**:
   - Since \( 3 \geq 1 \), we use the second piece of the function: \( f(x) = -x + 1 \).
   - Substituting \( x = 3 \):
   \[
   f(3) = -3 + 1 = -2
   \]

4. **\( f(6 + 1) \)**:
   - First, calculate \( 6 + 1 = 7 \). Since \( 7 \geq 1 \), we use the second piece of the function: \( f(x) = -x + 1 \).
   - Substituting \( x = 7 \):
   \[
   f(7) = -7 + 1 = -6
   \]

### Domain and Range

- **Domain**: The function \( f(x) \) is defined for all real numbers \( x \). Thus, the domain is \( (-\infty, \infty) \).
  
- **Range**: 
  - For \( x < 1 \), \( f(x) = x + 1 \), which covers the range \( (-\infty, 2) \).
  - For \( x \geq 1 \), \( f(x) = -x + 1 \), which covers the range \( (-\infty, 1] \).
  - Therefore, the overall range of \( f(x) \) is \( (-\infty, 2) \).

### Summary:
- \( f(-3) = -2 \)
- \( f(1) = 0 \)
- \( f(3) = -2 \)
- \( f(7) = -6 \)
- Domain: \( (-\infty, \infty) \)
- Range: \( (-\infty, 2) \)

Would you like more details on any part of this explanation, or do you have any additional questions?

### Related Questions:
1. How do you graph piecewise functions?
2. What are the steps to determine the range of a piecewise function?
3. How do you find where a piecewise function is continuous?
4. How do you determine the domain of more complex piecewise functions?
5. What is the difference between the domain and range of a function?
6. Can the range of a piecewise function include all real numbers?
7. How do you evaluate a piecewise function given a piece of the domain?
8. What happens to the function as \( x \) approaches 1 from the left and right?

### Tip:
When dealing with piecewise functions, carefully analyze each piece and the conditions attached to it to avoid any confusion during evaluation.

Learn how to evaluate a piecewise function and analyze its domain and range. This problem involves a piecewise function defined by cases, covering scenarios where different rules apply based on the value of x. Suitable for high school students studying piecewise functions and domain-range analysis.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation