Math Problem Statement
Sketch the graph of:
Sketch the graph of the piecewise function provided in the image.
Solution
The piecewise function given in the image is:
\begin{cases} 5, & x < -3 \\ x + 1, & -3 \leq x < 0 \\ -x^2 + 1, & x \geq 0 \end{cases}$$ ### Step-by-step breakdown: 1. **For $$x < -3$$:** $$f(x) = 5$$ This is a constant function. For values of $$x$$ less than $$-3$$, the graph will be a horizontal line at $$y = 5$$ up to $$x = -3$$, not including $$x = -3$$. 2. **For $$-3 \leq x < 0$$:** $$f(x) = x + 1$$ This is a linear function with a slope of 1 and a y-intercept of 1. The graph will be a straight line starting at $$x = -3$$, where $$f(-3) = -3 + 1 = -2$$, and ending just before $$x = 0$$, where $$f(0) = 0 + 1 = 1$$. The endpoint at $$x = 0$$ is excluded because of the strict inequality. 3. **For $$x \geq 0$$:** $$f(x) = -x^2 + 1$$ This is a downward-facing parabola with vertex at $$(0, 1)$$. The graph will start at $$(0, 1)$$ and follow the parabola downward as $$x$$ increases. ### Graph Sketch: 1. **For $$x < -3$$:** A horizontal line at $$y = 5$$ from $$x = -\infty$$ to just before $$x = -3$$. 2. **For $$-3 \leq x < 0$$:** A straight line from $$(-3, -2)$$ to $$(0, 1)$$ (but excluding $$x = 0$$). 3. **For $$x \geq 0$$:** A downward-facing parabola starting at $$(0, 1)$$ and curving downward. Would you like me to generate an image of the graph for you? Here are some related questions for deeper understanding: 1. What are the domain and range of the function $$f(x)$$? 2. How does the continuity of the function change at the boundaries $$x = -3$$ and $$x = 0$$? 3. What are the intercepts of the function with the axes? 4. Can you describe the behavior of the function as $$x \to -\infty$$ and $$x \to \infty$$? 5. What is the vertex of the parabola in the third case $$f(x) = -x^2 + 1$$? **Tip:** In piecewise functions, carefully evaluate the continuity at the boundary points by checking the left-hand and right-hand limits.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Linear Equations
Quadratic Equations
Parabolas
Formulas
f(x) = 5 (for x < -3)
f(x) = x + 1 (for -3 ≤ x < 0)
f(x) = -x^2 + 1 (for x ≥ 0)
Theorems
Continuity
Quadratic Function Properties
Parabola Vertex Formula
Suitable Grade Level
Grades 10-12
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