Question1. $f(x)=\left\{\begin{array}{l}x+3 \text { if } x \neq 3 \\ 2 \text { if } x=3\end{array}\right.$

Studdy Solution

STEP 1

1. We are given a piecewise function $f(x)$ .
2. The function is defined differently for $x = 3$ and $x \neq 3$ .
3. We need to analyze the behavior of the function at and around $x = 3$ .

STEP 2

1. Understand the piecewise function definition.
2. Evaluate the function for $x \neq 3$ .
3. Evaluate the function at $x = 3$ .
4. Analyze the continuity of the function at $x = 3$ .

STEP 3

The function $f(x)$ is defined piecewise. This means that its definition changes based on the value of $x$ . Specifically, it is defined as:
- $f(x) = x + 3$ when $x \neq 3$ - $f(x) = 2$ when $x = 3$

STEP 4

For $x \neq 3$ , the function is defined as $f(x) = x + 3$ . This is a linear function, and it is defined for all $x$ except $x = 3$ .

STEP 5

At $x = 3$ , the function is specifically defined as $f(x) = 2$ . This is a constant value at this single point.

STEP 6

To analyze the continuity at $x = 3$ , we need to check the left-hand limit, right-hand limit, and the value of the function at $x = 3$ .
- Left-hand limit as $x \to 3^-$ : $\lim_{x \to 3^-} f(x) = 3 + 3 = 6$ - Right-hand limit as $x \to 3^+$ : $\lim_{x \to 3^+} f(x) = 3 + 3 = 6$ - Function value at $x = 3$ : $f(3) = 2$
Since the left-hand limit and right-hand limit are equal but not equal to the function value at $x = 3$ , the function is not continuous at $x = 3$ .
The function $f(x)$ is continuous everywhere except at $x = 3$ , where it has a jump discontinuity.

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