Math Problem Statement
Sketch a graph of f(x), where:
- f(x) = 0 if x <= -1
- f(x) = -x - 1 if -1 < x <= 2
- f(x) = -2 if x > 2
Solution
The function is defined piecewise as follows:
\begin{cases} 0 & \text{if } x \leq -1 \\ -x - 1 & \text{if } -1 < x \leq 2 \\ -2 & \text{if } x > 2 \end{cases}$$ ### To sketch the graph of $$f(x)$$: 1. **For $$x \leq -1$$:** - $$f(x) = 0$$, which is a constant function. This corresponds to a horizontal line at $$y = 0$$ for all values of $$x \leq -1$$. - The point $$(-1, 0)$$ is included in this part of the function. 2. **For $$-1 < x \leq 2$$:** - $$f(x) = -x - 1$$, which is a linear function with slope $$-1$$ and y-intercept $$-1$$. The graph will be a straight line between $$x = -1$$ and $$x = 2$$. - At $$x = -1$$, the function gives $$f(-1) = -(-1) - 1 = 0$$, which matches the boundary point. - At $$x = 2$$, the function gives $$f(2) = -(2) - 1 = -3$$, so the point $$(2, -3)$$ is part of the graph. 3. **For $$x > 2$$:** - $$f(x) = -2$$, another constant function. This is a horizontal line at $$y = -2$$ for all $$x > 2$$. ### Key Points: - The graph starts at $$(-1, 0)$$ and remains at $$y = 0$$ for $$x \leq -1$$. - For $$-1 < x \leq 2$$, the function follows the line $$-x - 1$$, which passes through $$(-1, 0)$$ and $$(2, -3)$$. - For $$x > 2$$, the function becomes $$y = -2$$, so it is a flat line. Would you like me to create the graph for you? Also, here are some related questions: 1. What is the slope of the linear portion of $$f(x)$$ for $$-1 < x \leq 2$$? 2. How do we determine the continuity of piecewise functions at the points where the pieces change? 3. What are the limits of $$f(x)$$ as $$x \to -1^+$$ and $$x \to 2^-$$? 4. How does the behavior of $$f(x)$$ change as $$x \to \infty$$? 5. Can the function $$f(x)$$ be expressed as a combination of Heaviside step functions? **Tip:** When sketching piecewise functions, always check the values at the boundaries where the function definition changes.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise functions
Linear equations
Constant functions
Formulas
f(x) = -x - 1
f(x) = 0
f(x) = -2
Theorems
Continuity and Discontinuity in piecewise functions
Suitable Grade Level
Grades 9-12