https://file.aiquickdraw.com/mathbot/file/1729242833489ehw844jc.jpg

To solve this problem, we need to ensure that the function \( f(x) \) is continuous at \( x = 2 \).

The piecewise function is defined as:

\[
f(x) = 
\begin{cases} 
2x - 4 & \text{for } x < 2 \\
2 & \text{for } x = 2 \\
x - 2 & \text{for } x > 2
\end{cases}
\]

To remove the discontinuity at \( x = 2 \), the left-hand limit as \( x \to 2^- \) (from the left) must equal the right-hand limit as \( x \to 2^+ \) (from the right), and both must equal \( f(2) \).

### Step 1: Left-hand limit (\( x \to 2^- \))
For \( x < 2 \), the function is \( 2x - 4 \). As \( x \to 2 \) from the left:
\[
\lim_{x \to 2^-} f(x) = 2(2) - 4 = 4 - 4 = 0
\]

### Step 2: Right-hand limit (\( x \to 2^+ \))
For \( x > 2 \), the function is \( x - 2 \). As \( x \to 2 \) from the right:
\[
\lim_{x \to 2^+} f(x) = 2 - 2 = 0
\]

### Step 3: Adjust \( f(2) \)
To remove the discontinuity, \( f(2) \) should be equal to the limit from both sides. Since both the left-hand and right-hand limits at \( x = 2 \) are 0, we should set \( f(2) = 0 \).

Thus, the value of \( f(2) \) should be changed to \( 0 \) to remove the discontinuity.

### Answer: The correct choice is **D. 0**.

Would you like more details or have any questions?

---

Here are 5 related questions to expand on this topic:
1. What are the different types of discontinuities in a function?
2. How do you determine the continuity of a piecewise function in general?
3. Why is matching the left-hand and right-hand limits crucial for continuity?
4. How does the Intermediate Value Theorem relate to continuous functions?
5. Can a function be continuous at some points and discontinuous at others?

**Tip:** Always check the limits from both sides at the point of interest when analyzing continuity in piecewise functions.

To what new value should f(2) be changed to remove the discontinuity? The piecewise function is defined as: f(x) = { 2x - 4, if x < 2; 2, if x = 2; x - 2, if x > 2 }.

Learn how to determine the correct value of f(2) to remove the discontinuity in a piecewise function. This problem explains key concepts like limits and continuity, suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation