Math Problem Statement
Graph f(x)=-1/2x^2 if x<1 and 2x+1 if x≥1
Solution
To graph the piecewise function , which is defined as:
\begin{cases} -\frac{1}{2}x^2 & \text{if } x < 1 \\ 2x + 1 & \text{if } x \geq 1 \end{cases}$$ ### Steps to graph: 1. **For $$x < 1$$:** - The function is $$f(x) = -\frac{1}{2}x^2$$, which is a downward-facing parabola (since the coefficient of $$x^2$$ is negative). - Since we are only interested in values where $$x < 1$$, we graph this part from $$x = -\infty$$ to $$x = 1$$ (not including 1). - At $$x = 1$$, we calculate $$f(1) = -\frac{1}{2}(1)^2 = -\frac{1}{2}$$. So, the left side of the graph stops just before the point $$(1, -\frac{1}{2})$$. 2. **For $$x \geq 1$$:** - The function is $$f(x) = 2x + 1$$, which is a straight line with a slope of 2 and a y-intercept of 1. - Starting from $$x = 1$$, the point at $$x = 1$$ is $$f(1) = 2(1) + 1 = 3$$, so the graph starts at $$(1, 3)$$ (inclusive) and continues upward for $$x \geq 1$$. ### Key Features: - The graph is composed of two distinct parts: - A downward-facing parabola for $$x < 1$$. - A straight line for $$x \geq 1$$. - At $$x = 1$$, there is a **jump discontinuity** since $$f(x)$$ changes from $$-\frac{1}{2}$$ (approaching from the left) to 3 (approaching from the right). Would you like a visual graph or additional details on how to sketch it? Here are some related questions: 1. How do you find the domain and range of this piecewise function? 2. What is the derivative of each part of this piecewise function? 3. How does the discontinuity at $$x = 1$$ affect the behavior of the graph? 4. How can you determine if a piecewise function is continuous? 5. What happens if we change the point of discontinuity in this piecewise function? **Tip:** When dealing with piecewise functions, always pay close attention to how each piece is defined, especially at the boundaries where the pieces meet.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Quadratic Functions
Linear Functions
Graphing
Discontinuity
Formulas
f(x) = -1/2x^2 for x < 1
f(x) = 2x + 1 for x ≥ 1
Theorems
Discontinuity Theorem
Quadratic Function Properties
Linear Function Properties
Suitable Grade Level
Grades 10-12