https://file.aiquickdraw.com/mathbot/file/1726784323739nz9165vw.jpg

Let's go through the problem step by step.

The function \( f(x) \) is defined in a piecewise manner:

\[
f(x) = 
\begin{cases}
2 + x^2 & \text{if } x \leq 0, \\
4 - x & \text{if } 0 < x \leq 4, \\
(x - 4)^2 & \text{if } x > 4.
\end{cases}
\]

### Part (a): Find the x-value at which \( f \) is discontinuous.

To find where the function might be discontinuous, we need to check the points where the pieces of the function switch, i.e., at \( x = 0 \) and \( x = 4 \).

#### Checking \( x = 0 \):
- **Left-hand limit** as \( x \to 0^- \): 
  \[
  \lim_{x \to 0^-} f(x) = 2 + (0)^2 = 2.
  \]
  
- **Right-hand limit** as \( x \to 0^+ \):
  \[
  \lim_{x \to 0^+} f(x) = 4 - 0 = 4.
  \]
  
Since the left-hand and right-hand limits are different, \( f(x) \) is **discontinuous at \( x = 0 \)**.

#### Checking \( x = 4 \):
- **Left-hand limit** as \( x \to 4^- \):
  \[
  \lim_{x \to 4^-} f(x) = 4 - 4 = 0.
  \]
  
- **Right-hand limit** as \( x \to 4^+ \):
  \[
  \lim_{x \to 4^+} f(x) = (4 - 4)^2 = 0.
  \]

Since the left-hand and right-hand limits are equal and equal to \( f(4) = 0 \), the function is **continuous at \( x = 4 \)**.

Thus, \( f(x) \) is only discontinuous at \( x = 0 \).

### Part (b): Determine whether \( f \) is continuous from the right, from the left, or neither at \( x = 0 \).

- From part (a), we found that the left-hand limit at \( x = 0 \) is 2, and the right-hand limit is 4. Since these limits do not match, \( f(x) \) is neither continuous from the right nor from the left. The correct answer should be "**neither**."

It seems the selected option in the image ("continuous from the right") is incorrect.

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Would you like more details on the continuity concepts or any other part? Here are a few related questions you might find helpful:

1. How do you determine the left-hand and right-hand limits for piecewise functions?
2. What does it mean for a function to be continuous from the left or the right?
3. Can a function be discontinuous at one point but still continuous elsewhere?
4. What types of discontinuities can a function have (e.g., jump, infinite, removable)?
5. How can you graphically represent the continuity of a piecewise function?

**Tip:** When working with piecewise functions, always check for continuity at the points where the pieces switch by comparing the left-hand and right-hand limits.

Find the x-value at which f is discontinuous and determine whether f is continuous from the right, from the left, or neither.

Determine the point of discontinuity of a piecewise function and whether it is continuous from the right, from the left, or neither at x = 0. This problem covers key calculus concepts such as limits, continuity, and piecewise functions. Suitable for students in Grades 10-12.

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