Math Problem Statement
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).
Solution
The problem you've uploaded asks you to graph the piecewise function , 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:
\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.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Graphing
Function Evaluation
Formulas
f(x) = -2x + 1 for x ≤ 2
f(x) = 5x - 4 for x > 2
Theorems
Vertical Line Test for Functions
Suitable Grade Level
Grades 8-10