Math Problem Statement
Evaluate the function when x = -2, 0, and 1/2 for the piecewise function f(x) = {-2x + 1, if x ≤ 0; x - 3, if x > 0}. Show your work.
Solution
The function provided is:
-2x + 1 & \text{if } x \leq 0 \\ x - 3 & \text{if } x > 0 \end{cases}$$ We are asked to evaluate the function at three points: $$x = -2$$, $$x = 0$$, and $$x = \frac{1}{2}$$. 1. **For $$x = -2$$:** Since $$-2 \leq 0$$, we use the first case: $$f(x) = -2x + 1$$. $$f(-2) = -2(-2) + 1 = 4 + 1 = 5$$ 2. **For $$x = 0$$:** Since $$0 \leq 0$$, we again use the first case: $$f(x) = -2x + 1$$. $$f(0) = -2(0) + 1 = 0 + 1 = 1$$ 3. **For $$x = \frac{1}{2}$$:** Since $$\frac{1}{2} > 0$$, we use the second case: $$f(x) = x - 3$$. $$f\left(\frac{1}{2}\right) = \frac{1}{2} - 3 = \frac{1}{2} - \frac{6}{2} = -\frac{5}{2}$$ Thus, the values are: - $$f(-2) = 5$$ - $$f(0) = 1$$ - $$f\left(\frac{1}{2}\right) = -\frac{5}{2}$$ Would you like more details or have any questions? Here are 5 related questions to expand your understanding: 1. How do we determine which piece of the function to use? 2. What would happen if the function had more than two cases? 3. How do you handle fractions or decimals in piecewise functions? 4. What is the significance of the boundary condition $$x = 0$$? 5. How would this function behave as $$x$$ approaches infinity? **Tip:** Always check the conditions for a piecewise function carefully to ensure you're using the correct part of the function.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Algebra
Formulas
f(x) = -2x + 1 for x ≤ 0
f(x) = x - 3 for x > 0
Theorems
Piecewise Function Evaluation
Suitable Grade Level
Grades 8-10