Identify all singular points and points of discontinuity of the given function. HINT [See Example 3.] (Enter your answers as comma-separated lists. If an answer does not exist, enter DNE.)

*f*(*x*) = 

2*x* − 9    

if *x* < 5

−2*x* + 11    

if *x* > 5

The given function is a piecewise function defined as:

\[
f(x) =
\begin{cases}
2x - 9 & \text{if } x < 5 \\
-2x + 11 & \text{if } x > 5
\end{cases}
\]

### Step 1: Check for continuity at \( x = 5 \)

To identify points of discontinuity, we check if the left-hand limit (as \( x \to 5^- \)) and the right-hand limit (as \( x \to 5^+ \)) are equal to each other and to the value of the function at \( x = 5 \).

1. **Left-hand limit** as \( x \to 5^- \):
   - From the first part of the function, \( f(x) = 2x - 9 \).
   - \( \lim_{x \to 5^-} f(x) = 2(5) - 9 = 10 - 9 = 1 \).

2. **Right-hand limit** as \( x \to 5^+ \):
   - From the second part of the function, \( f(x) = -2x + 11 \).
   - \( \lim_{x \to 5^+} f(x) = -2(5) + 11 = -10 + 11 = 1 \).

Since both the left-hand and right-hand limits are equal to 1, the function is continuous at \( x = 5 \). Therefore, **there is no discontinuity at \( x = 5 \)**.

### Step 2: Check for singular points

Singular points occur where the function is undefined or where there are points of discontinuity. Since the function is defined piecewise and is continuous at \( x = 5 \), there are **no singular points**.

### Final Answer:
- **Singular points**: DNE
- **Points of discontinuity**: DNE

Would you like further details or have any questions?

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Here are 5 related questions to expand on this topic:

1. How do you determine the left-hand and right-hand limits of a function at a point?
2. What are the conditions for a function to be continuous at a point?
3. Can a function be continuous but still have singular points?
4. How would you handle a piecewise function with more than two cases?
5. What types of discontinuities exist, and how are they classified?

**Tip:** Always check both the limits and the function value at a point to determine continuity!

Identify all singular points and points of discontinuity of the given function. HINT [See Example 3.] (Enter your answers as comma-separated lists. If an answer does not exist, enter DNE.)

f(x) = 

2x − 9 if x < 5

−2x + 11 if x > 5

Learn how to identify singular points and discontinuities in a piecewise function with a step-by-step analysis of limits and continuity at x = 5. This problem explores key concepts like piecewise functions, limits, and continuity, suitable for students in Grades 10-12.

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