Math Problem Statement
Suppose that the function f(x) is defined as a piecewise function with f(x) = -x + 3 for x <= -1 and f(x) = -3x + 1 for x >= -1.
Solution
The piecewise function in the image is defined as follows:
\begin{cases} -x + 3 & \text{if } x \leq -1 \\ -3x + 1 & \text{if } x \geq -1 \end{cases}$$ ### To graph this function: 1. **For $$x \leq -1$$**, use the equation $$f(x) = -x + 3$$, which represents a straight line with slope $$-1$$ and a y-intercept at 3. 2. **For $$x \geq -1$$**, use the equation $$f(x) = -3x + 1$$, which represents another straight line with slope $$-3$$ and y-intercept at 1. Both of these pieces meet at $$x = -1$$, so we should check the values of the function at this point to ensure continuity. - For $$x = -1$$ in the first piece, $$f(-1) = -(-1) + 3 = 1 + 3 = 4$$. - For $$x = -1$$ in the second piece, $$f(-1) = -3(-1) + 1 = 3 + 1 = 4$$. Since the function equals 4 at $$x = -1$$ for both pieces, the function is continuous at that point. Would you like further details or any specific clarifications? ### Here are 5 related questions to consider: 1. How do you find the slope and intercept of a linear equation? 2. What does it mean for a function to be continuous? 3. How do you determine if a piecewise function is differentiable at a point? 4. How do you solve for the domain and range of a piecewise function? 5. Can you find the intersection points of the two parts of the piecewise function? ### Tip: When graphing piecewise functions, it's important to plot key points where the function switches from one rule to another, and check for continuity or jumps at those points.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Linear Functions
Continuity
Formulas
f(x) = -x + 3 for x <= -1
f(x) = -3x + 1 for x >= -1
Theorems
Continuity at x = -1
Suitable Grade Level
Grades 9-12