https://file.aiquickdraw.com/mathbot/file/1726712812974qcftebug.jpg

The problem you've uploaded asks you to graph the piecewise function \( f(x) \), determine if it's a function, and evaluate it at specific domain values. Let's break this down step by step.

The function is defined as:

\[
f(x) = 
\begin{cases} 
-2x + 1 & \text{for } x \leq 2 \\
5x - 4 & \text{for } x > 2 
\end{cases}
\]

### Step 1: Graph the Piecewise Function
- **For \( x \leq 2 \)**, you need to graph the line \( f(x) = -2x + 1 \).
- **For \( x > 2 \)**, you graph the line \( f(x) = 5x - 4 \).

### Step 2: Identify if it's a Function
A graph represents a function if each input (x-value) has exactly one output (y-value). This is confirmed if the graph passes the vertical line test, meaning that no vertical line intersects the graph at more than one point.

### Step 3: Evaluate the Function at Specific Domain Values
You are asked to find the function's value at three points:

1. **\( f(-4) \)**:
   Since \( -4 \leq 2 \), use the equation \( f(x) = -2x + 1 \) for this domain.
   \[
   f(-4) = -2(-4) + 1 = 8 + 1 = 9
   \]

2. **\( f(8) \)**:
   Since \( 8 > 2 \), use the equation \( f(x) = 5x - 4 \) for this domain.
   \[
   f(8) = 5(8) - 4 = 40 - 4 = 36
   \]

3. **\( f(2) \)**:
   Since \( 2 \leq 2 \), use the equation \( f(x) = -2x + 1 \) for this domain.
   \[
   f(2) = -2(2) + 1 = -4 + 1 = -3
   \]

### Conclusion:
1. The function is **yes**, a function because each x-value corresponds to exactly one y-value.
2. The evaluations are:
   - \( f(-4) = 9 \)
   - \( f(8) = 36 \)
   - \( f(2) = -3 \)

Would you like a more detailed explanation or a graph? 

Here are some related questions to further explore:
1. How do you know if a piecewise function is continuous?
2. What is the significance of the vertical line test?
3. How can we determine the domain and range of a piecewise function?
4. Can a piecewise function have a discontinuity at a point?
5. How would the function change if the inequalities for \(x\) were reversed?

**Tip:** When working with piecewise functions, always pay close attention to the domain restrictions for each part of the function.

Carefully graph each of the following. Identify whether or not the graph is a function. Then, evaluate the graph at any specified domain value. f(x) = -2x + 1 for x ≤ 2, 5x - 4 for x > 2. Evaluate at f(-4), f(8), and f(2).

Learn how to graph and evaluate the piecewise function f(x) = -2x + 1 for x ≤ 2 and f(x) = 5x - 4 for x > 2. This tutorial also demonstrates how to determine if the graph represents a function using the vertical line test, and evaluates the function at f(-4), f(8), and f(2). Suitable for Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation