Math Problem Statement
Based on the graph, find the range of y = f(x).
Solution
The question asks for the range of the function , which is defined piecewise, as seen in the graph. Let's analyze the function and the corresponding graph step by step.
The function is given as a piecewise function:
\begin{cases} 4 & \text{if } -6 \leq x \leq -2 \\ |x| & \text{if } -2 \leq x < 8 \\ \sqrt{x} & \text{if } 8 \leq x \leq 13 \end{cases}$$ ### Step-by-step analysis: 1. **For the interval** $$-6 \leq x \leq -2$$, $$f(x) = 4$$. - This is a constant function, so the value of $$f(x)$$ is 4 over this entire interval. 2. **For the interval** $$-2 \leq x < 8$$, $$f(x) = |x|$$. - Here, $$f(x)$$ takes values from $$|x| = |-2| = 2$$ to $$|x| = 8$$ (as the absolute value increases from 2 to 8 over the range). 3. **For the interval** $$8 \leq x \leq 13$$, $$f(x) = \sqrt{x}$$. - In this case, $$f(x)$$ takes values from $$\sqrt{8} \approx 2.83$$ to $$\sqrt{13} \approx 3.61$$. ### Finding the range of $$f(x)$$: From the graph and the piecewise function, we can now determine the range: - The function takes the constant value 4 in the interval $$-6 \leq x \leq -2$$. - Then, from $$-2 \leq x < 8$$, the function takes values from $$2$$ to $$8$$ (due to $$|x|$$). - Finally, in the interval $$8 \leq x \leq 13$$, the function's values range from $$\sqrt{8} \approx 2.83$$ to $$\sqrt{13} \approx 3.61$$. Thus, the complete range of $$f(x)$$ is $$[0, 8]$$, as the function covers all values from 0 (though it never actually reaches 0 but approaches it) to 8 across the various intervals. ### Correct Answer: $$[0, 8]$$ Let me know if you need further details or clarifications! Here are five follow-up questions related to piecewise functions and range concepts: 1. What is the definition of a piecewise function? 2. How do you find the domain of a piecewise function? 3. Why is the square root function only defined for non-negative values of $$x$$? 4. How can absolute value functions affect the range of a function? 5. What are some common strategies for graphing piecewise functions? **Tip:** When analyzing a piecewise function, always look at each segment individually to determine its effect on the overall range and domain!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Absolute Value
Square Root Functions
Range of Functions
Formulas
Absolute Value: |x|
Square Root Function: √x
Theorems
Range of a Function
Properties of Absolute Value
Properties of Square Root Function
Suitable Grade Level
Grades 10-12