Math Problem Statement
Solve
Evaluate f(-2), f(0), f(1) and find the domain of the piecewise-defined function f(x) = {x if x >= 0, -x if x < 0}.
Solution
We are given a piecewise-defined function defined as:
\begin{cases} x, & \text{if } x \geq 0 \\ -x, & \text{if } x < 0 \end{cases}$$ ### 1. **Evaluating the function:** Let's evaluate the function at the specified points: - $$f(-2)$$: Since $$-2 < 0$$, we use the second case, $$f(x) = -x$$. $$f(-2) = -(-2) = 2$$ - $$f(0)$$: Since $$0 \geq 0$$, we use the first case, $$f(x) = x$$. $$f(0) = 0$$ - $$f(1)$$: Since $$1 \geq 0$$, we use the first case, $$f(x) = x$$. $$f(1) = 1$$ ### 2. **Finding the domain of $$f(x)$$:** The domain of $$f(x)$$ is the set of all values of $$x$$ for which the function is defined. Since the function provides a rule for both $$x \geq 0$$ and $$x < 0$$, the domain of $$f(x)$$ is all real numbers: $$\text{Domain of } f(x): (-\infty, \infty)$$ ### 3. **Sketching the graph:** The function behaves as: - $$f(x) = x$$ for $$x \geq 0$$, which is a line with a slope of 1 passing through the origin. - $$f(x) = -x$$ for $$x < 0$$, which is a line with a slope of -1, also passing through the origin. At $$x = 0$$, the value is $$f(0) = 0$$, so the graph is continuous at this point. Would you like me to sketch the graph as well? --- Here are some related questions for further practice: 1. What is the range of $$f(x)$$? 2. Is $$f(x)$$ continuous at all points? 3. Find the inverse of $$f(x)$$. 4. Is the function $$f(x)$$ differentiable at $$x = 0$$? 5. How does the graph of $$f(x)$$ change if we modify it to $$f(x) = 2x$$ when $$x \geq 0$$? **Tip:** For piecewise functions, always check which condition applies to a specific input before evaluating it.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Algebra
Function Evaluation
Formulas
f(x) = x for x >= 0
f(x) = -x for x < 0
Theorems
Piecewise function evaluation and continuity at x = 0
Suitable Grade Level
Grades 9-12