Math Problem Statement

Graph the piecewise-defined function. Use the graph to determine the domain and range of the function: f(x) = { 2x - 2, if x ≤ 0 -x + 1, if x > 0 } Choose the correct graph and determine the domain and range.

Solution

The piecewise-defined function is given as:

\begin{cases} 2x - 2 & \text{if } x \leq 0 \\ -x + 1 & \text{if } x > 0 \end{cases}$$ Let's break down what this means in terms of the graph: 1. **For $$x \leq 0$$**: The function follows the equation $$f(x) = 2x - 2$$. This is a linear equation with a slope of 2 and a y-intercept of $$-2$$. The graph for this part will be a line extending from $$x = 0$$ and continuing to the left. 2. **For $$x > 0$$**: The function follows the equation $$f(x) = -x + 1$$. This is another linear equation, with a slope of $$-1$$ and a y-intercept of 1. The graph for this part will be a line starting just to the right of $$x = 0$$ and extending to the right. ### Identifying the Correct Graph From the options provided, the correct graph will show: - A line with slope 2 for $$x \leq 0$$, starting from $$y = -2$$ at $$x = 0$$. - A line with slope $$-1$$ for $$x > 0$$, starting from $$y = 1$$ at $$x = 0$$, but the point $$x = 0$$ should be an open circle (because the function for $$x = 0$$ is defined by the left-hand side of the piecewise function). ### Determining the Domain and Range - **Domain**: The domain includes all possible $$x$$-values where the function is defined. Since the function is defined for all real $$x$$ (both positive and negative values), the domain is: $$(-\infty, \infty)$$ - **Range**: The range is the set of all possible $$y$$-values. We analyze the behavior of each part of the piecewise function: - For $$x \leq 0$$, the line $$y = 2x - 2$$ can take any value from $$-\infty$$ to $$0$$ (including $$0$$). - For $$x > 0$$, the line $$y = -x + 1$$ decreases from 1 (but does not include 1, due to the open circle at $$x = 0$$) to $$-\infty$$. Therefore, the range is: $$(-\infty, 1)$$ ### Answer Choices - The correct graph would be **option C**. - Domain: $$(-\infty, \infty)$$ - Range: $$(-\infty, 1)$$ Would you like to go over any details or need further explanations? Here are five related questions: 1. How do you determine the slope and intercept from a linear equation? 2. What is the significance of open and closed circles in piecewise functions? 3. Can piecewise functions have discontinuities, and if so, how do you identify them? 4. How would you solve for specific points of intersection between two linear functions in a piecewise function? 5. How does changing the slope of a line affect its graph? **Tip**: When working with piecewise functions, carefully check the boundaries (like $$x = 0$$) to ensure you're applying the correct part of the function.

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Math Problem Analysis

Mathematical Concepts

Piecewise Functions
Linear Equations
Domain and Range

Formulas

f(x) = 2x - 2 for x ≤ 0
f(x) = -x + 1 for x > 0

Theorems

Domain and Range of Piecewise Functions

Suitable Grade Level

Grades 10-12

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