Math Problem Statement

The domain of the piecewise function is (-∞, ∞). Graph the function and use the graph to determine the function's range. The function is defined as: f(x) = -x if x < 0, f(x) = x if x ≥ 0. Choose the correct graph and determine the range.

Solution

The piecewise function given is:

\begin{cases} -x & \text{if } x < 0 \\ x & \text{if } x \geq 0 \end{cases}$$ ### Part a: Graph the function For $$f(x) = -x$$ when $$x < 0$$, the graph will be a line with a negative slope of $$-1$$ on the left-hand side (when $$x < 0$$). For $$f(x) = x$$ when $$x \geq 0$$, the graph will be a line with a positive slope of $$1$$ on the right-hand side (when $$x \geq 0$$). The two pieces meet at $$x = 0$$, where the value of $$f(x)$$ is 0. This results in a "V" shape. The correct graph is **C**. --- ### Part b: Range of the function From the graph, we can see that the function covers all possible $$y$$-values as $$x$$ moves across the real numbers. Therefore, the range of the function is: $$\text{Range: } (-\infty, \infty)$$ Let me know if you want further details or explanations! --- Here are 5 follow-up questions to expand your understanding: 1. What does the domain of a function represent, and how is it determined? 2. How does the slope of the lines in each piece relate to the definition of the function? 3. What would happen to the graph if the conditions for $$x$$ were swapped? 4. Can a piecewise function have a discontinuous graph? Why or why not? 5. How can the range of a piecewise function differ from its domain? **Tip:** When graphing piecewise functions, always check the boundaries (e.g., $$x = 0$$ here) to ensure the pieces connect properly.

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Math Problem Analysis

Mathematical Concepts

Piecewise Functions
Graphing Functions
Function Range

Formulas

f(x) = -x if x < 0
f(x) = x if x ≥ 0

Theorems

Properties of Linear Functions
Domain and Range of Functions

Suitable Grade Level

Grades 9-11