Math Problem Statement
Graph the piecewise function f(x) = { -x + 3 if x < 2, 2x - 3 if x >= 2 }, and determine which option correctly represents the function.
Solution
To determine which graph correctly represents the piecewise function , let us analyze the given function:
\begin{cases} -x + 3 & \text{if } x < 2 \\ 2x - 3 & \text{if } x \geq 2 \end{cases}$$ ### Step-by-Step Analysis: 1. **For $$x < 2$$:** The function is $$f(x) = -x + 3$$, which is a linear equation with: - Slope = $$-1$$ - Y-intercept = $$3$$. At $$x = 2$$ (excluded), the value is: $$f(2^-) = -(2) + 3 = 1$$ So, the graph will have an open circle at $$(2, 1)$$. 2. **For $$x \geq 2$$:** The function is $$f(x) = 2x - 3$$, which is a linear equation with: - Slope = $$2$$ - Y-intercept = $$-3$$. At $$x = 2$$ (included), the value is: $$f(2) = 2(2) - 3 = 1$$ The graph will start with a filled circle at $$(2, 1)$$. ### Key Features of the Graph: - The line $$-x + 3$$ continues for $$x < 2$$. - The line $$2x - 3$$ begins at $$(2, 1)$$ and continues for $$x \geq 2$$. ### Observing the Options: - **Option A** correctly shows: - $$f(x) = -x + 3$$ for $$x < 2$$ with an open circle at $$(2, 1)$$, - $$f(x) = 2x - 3$$ for $$x \geq 2$$ with a filled circle at $$(2, 1)$$. Thus, the correct answer is **Option A**. --- Would you like a detailed explanation of graphing piecewise functions? Here are five questions for further exploration: 1. How do you find the domain and range of a piecewise function? 2. What role do open and closed circles play in graphing? 3. How do you calculate the slope and y-intercept of linear equations? 4. What happens at the boundary point of a piecewise function? 5. Can piecewise functions be nonlinear? How would you graph them? **Tip:** When graphing piecewise functions, evaluate the function carefully at the boundaries and use open or closed circles to show inclusivity.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Linear Equations
Graphing
Formulas
f(x) = -x + 3 for x < 2
f(x) = 2x - 3 for x >= 2
Theorems
Definition of a Piecewise Function
Suitable Grade Level
Grades 9-12