Discontinuity of Piecewise Functions at x = 2

Math Problem Statement

Given the piecewise function: f(x) = { x^2 for x <= 2, 4 + x for x > 2 }, find the value of x at which the function is discontinuous.

The image you uploaded shows a mathematical function $f(x)$ defined piecewise:

x^2 & \text{for } x \leq 2 \\ 4 + x & \text{for } x > 2 \end{cases}$$ The question asks for the value of $$x$$ at which the function is discontinuous. To determine where the function might be discontinuous, we need to check the point where the piecewise parts meet, which is at $$x = 2$$. 1. **For $$x \leq 2$$**: The function is defined as $$f(x) = x^2$$, so $$f(2) = 2^2 = 4$$. 2. **For $$x > 2$$**: The function is defined as $$f(x) = 4 + x$$, so at $$x = 2$$, $$f(2^+) = 4 + 2 = 6$$. Since the values of $$f(2)$$ and $$f(2^+)$$ are different, the function is discontinuous at $$x = 2$$. Therefore, the correct answer is: **X = 2** Would you like a more detailed explanation or help with another question? Here are some related questions for further exploration: 1. What is the definition of a discontinuous function? 2. How can you identify continuity using limits? 3. Can piecewise functions be continuous at the junctions? 4. What are the different types of discontinuities in functions? 5. How would you find the limit of a function at a given point? **Tip:** Always check the function values on both sides of the point where the piecewise functions meet to determine continuity.

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Piecewise Functions
Continuity and Discontinuity
Limits

f(x) = x^2 for x <= 2
f(x) = 4 + x for x > 2

Continuity at a point
Discontinuity of piecewise functions

Grades 9-11

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