Math Problem Statement
The domain of the piecewise function is (-∞, ∞). Graph the function and determine the function's range based on the graph.
Solution
Let's analyze the problem step by step.
The function given is a piecewise function:
\begin{cases} -4x & \text{if } x < 0 \\ 4x & \text{if } x \geq 0 \end{cases}$$ ### (a) Graphing the function: - For $$x < 0$$, the function is $$f(x) = -4x$$, which is a straight line with a negative slope, passing through the origin. - For $$x \geq 0$$, the function is $$f(x) = 4x$$, which is a straight line with a positive slope, also passing through the origin. - Therefore, the graph of this function consists of two lines meeting at the origin, one with a negative slope for negative $$x$$, and one with a positive slope for positive $$x$$. In the image, option A shows this exact behavior: - A line with a negative slope on the left side (for $$x < 0$$), - A line with a positive slope on the right side (for $$x \geq 0$$). ### (b) Finding the range: The range refers to all possible $$y$$-values of the function. - As $$x$$ approaches negative infinity, $$y = -4x$$ grows positively without bound. - As $$x$$ approaches positive infinity, $$y = 4x$$ also grows positively without bound. - Therefore, the range of the function is all real numbers $$(- \infty, \infty)$$. Based on this analysis, the correct choice for the range is: - **A. The range does not have any isolated values. It can be described by $$(- \infty, \infty)$$**. --- Would you like a more detailed explanation or have any further questions? ### Related Questions: 1. What is the domain of a piecewise function? 2. How do you determine the slope of a line in a piecewise function? 3. What are common shapes for the graph of linear piecewise functions? 4. How does the behavior of the function at the boundary $$x = 0$$ affect the graph? 5. How do you describe the range of a function that includes only certain values? ### Tip: When graphing piecewise functions, always check for continuity at the boundary points to see if the function changes smoothly.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Graphing Linear Equations
Range of a Function
Formulas
f(x) = -4x if x < 0
f(x) = 4x if x >= 0
Theorems
None, basic graphing of linear functions
Suitable Grade Level
Grades 9-12
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