Math Problem Statement
What is the range of the piecewise function: f(x) = { 4x if x ≤ 0, 4 if x > 0 }? What graph represents the function, and how is the range calculated?
Solution
The piecewise function provided in the image is defined as:
\begin{cases} 4x & \text{if } x \leq 0 \\ 4 & \text{if } x > 0 \end{cases}$$ Let's analyze the function and the related graph to determine its range. ### Understanding the function: 1. **For $$x \leq 0$$:** The function behaves as $$f(x) = 4x$$, which is a linear equation. So, for negative or zero values of $$x$$, the function has a continuous, decreasing linear behavior. 2. **For $$x > 0$$:** The function is constant at $$f(x) = 4$$. So, for positive values of $$x$$, the function is a horizontal line at $$y = 4$$. ### Graph and Range: From the provided graph in the image (Graph B is selected), we can see: - The graph for $$x \leq 0$$ starts at 0 and moves downward. - For $$x > 0$$, the graph is flat at $$y = 4$$. #### Range Calculation: - For $$x \leq 0$$, since the function is linear with a slope of 4, the function values decrease without bound, i.e., as $$x$$ goes to negative infinity, $$f(x)$$ approaches negative infinity. - For $$x > 0$$, the function value is exactly 4. Hence, the range of the function is: $$(-\infty, 0] \cup \{4\}$$ This indicates that the function can take any value from negative infinity up to 0, and it also takes the isolated value of 4 when $$x > 0$$. ### Conclusion: - The graph corresponding to this function is **B**. - The range is: $$(- \infty, 0] \cup \{4\}$$. --- Would you like more details or have any further questions? Here are some related questions for further exploration: 1. How do you determine the range of a function from its graph? 2. What is the difference between continuous and discrete ranges? 3. How does the slope affect the graph of linear functions like $$4x$$? 4. Why is there an isolated value in the range for this piecewise function? 5. How do piecewise functions model real-world situations? **Tip:** Always check for both continuous intervals and isolated points when determining the range of piecewise functions.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Range of Functions
Graph Analysis
Formulas
f(x) = 4x if x ≤ 0
f(x) = 4 if x > 0
Theorems
Range of a Function
Suitable Grade Level
Grades 9-11