Math Problem Statement
Graph the function f(x) defined as: f(x) = {x - 4 if x < 2, -2x + 2 if x >= 2}. Determine whether the function is continuous.
Solution
The function given in the problem is a piecewise-defined function:
x - 4 & \text{if } x < 2 \\ -2x + 2 & \text{if } x \geq 2 \end{cases}$$ ### Analysis: 1. **Graph each piece**: - For $$x < 2$$, the function is $$f(x) = x - 4$$, which is a line with slope 1 and y-intercept -4. This segment is defined for $$x < 2$$. - For $$x \geq 2$$, the function is $$f(x) = -2x + 2$$, which is a line with slope -2 and y-intercept 2. This segment is defined for $$x \geq 2$$. 2. **Check for continuity at $$x = 2$$**: - **Left-hand limit**: $$\lim_{x \to 2^-} f(x) = 2 - 4 = -2$$. - **Right-hand limit**: $$\lim_{x \to 2^+} f(x) = -2(2) + 2 = -4 + 2 = -2$$. - **Function value at $$x = 2$$**: $$f(2) = -2(2) + 2 = -4 + 2 = -2$$. Since the left-hand limit, right-hand limit, and function value at $$x = 2$$ are all equal to -2, the function is **continuous** at $$x = 2$$. ### Conclusion: The function is continuous for all real numbers. Thus, the answer to the question is: **Yes, the function is continuous.** Would you like a more detailed explanation or have any questions? ### Related Questions: 1. How would you find the derivative of this piecewise function? 2. What happens if the slopes of the two pieces are equal? 3. How can you determine the continuity of a function graphically? 4. Can a piecewise function have a jump discontinuity? If so, how can you identify it? 5. What is the difference between pointwise and interval continuity? ### Tip: When analyzing piecewise functions for continuity, always check the left-hand and right-hand limits at the boundary points.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Continuity
Graphing
Formulas
f(x) = x - 4 if x < 2
f(x) = -2x + 2 if x >= 2
Theorems
Continuity at a Point
Suitable Grade Level
High School (Grades 9-12)