Math Problem Statement
f(x) is defined as:
f(x) =
5, if x < -1
|x|, if -1 ≤ x < 2
-6, if x ≥ 2
a. Find f(5), f(1), and f(-7).
b. Sketch the graph of the piecewise-defined function.
c. Determine the domain of f.
d. Determine the range of f.
Solution
The problem in your image involves a piecewise function defined as follows:
5 & \text{if } x < -1 \\ |x| & \text{if } -1 \leq x < 2 \\ -6 & \text{if } x \geq 2 \end{cases}$$ Let's break down each part of the problem: ### Part (a): Find $$f(5), f(1), \text{and } f(-7)$$ 1. **$$f(5)$$:** Since $$x = 5 \geq 2$$, we use the third piece of the function: $$f(5) = -6$$ 2. **$$f(1)$$:** Since $$-1 \leq 1 < 2$$, we use the second piece: $$f(1) = |1| = 1$$ 3. **$$f(-7)$$:** Since $$-7 < -1$$, we use the first piece: $$f(-7) = 5$$ So, the values are: $$f(5) = -6, \quad f(1) = 1, \quad f(-7) = 5$$ ### Part (b): Sketch the graph of the piecewise function To sketch the graph: - For $$x < -1$$, the function is constant at $$f(x) = 5$$, so it's a horizontal line at $$y = 5$$ for $$x < -1$$. - For $$-1 \leq x < 2$$, the function is $$f(x) = |x|$$, so it follows the absolute value shape, with a "V" starting at $$x = -1$$ up to $$x = 2$$. - For $$x \geq 2$$, the function is constant again at $$f(x) = -6$$, so it is a horizontal line at $$y = -6$$. ### Part (c): Determine the domain of $$f$$ The domain of $$f(x)$$ includes all real numbers because each piece of the function covers a part of the real line without any gaps. Therefore, the domain is: $$\boxed{(-\infty, \infty)}$$ ### Part (d): Determine the range of $$f$$ The range is the set of all possible values that the function can take: - The first piece, $$f(x) = 5$$, gives the value 5 for $$x < -1$$. - The second piece, $$f(x) = |x|$$, gives values between $$0$$ and $$1$$ for $$-1 \leq x < 2$$. - The third piece, $$f(x) = -6$$, gives the value -6 for $$x \geq 2$$. Therefore, the range is: $$\boxed{[-6, 5]}$$ Let me know if you need further clarification or more details on any part! --- ### Related Questions: 1. How would the graph of $$f(x)$$ change if the absolute value function was replaced with $$x^2$$? 2. What is the significance of the domain in a real-world application of a piecewise function? 3. How does the piecewise function's range change if the last condition is modified to $$f(x) = -x$$ for $$x \geq 2$$? 4. Can a piecewise function have a discontinuous domain, and how would you represent it? 5. How can you find the limits of $$f(x)$$ as $$x$$ approaches -1 and 2 from both sides? ### Tip: When working with piecewise functions, always pay attention to the conditions on $$x$$ to correctly evaluate and sketch each segment of the function!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Domain and Range
Absolute Value Functions
Formulas
f(x) = |x| for the second interval of the piecewise function
Theorems
Piecewise function continuity and evaluation
Suitable Grade Level
College Algebra or High School Advanced Algebra
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