Math Problem Statement
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
Solution
The given function is a piecewise function defined as:
\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? --- 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!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Limits
Continuity
Singular Points
Formulas
Left-hand limit: lim(x -> c-) f(x)
Right-hand limit: lim(x -> c+) f(x)
Theorems
Continuity at a Point Theorem
Limit Existence Criterion
Suitable Grade Level
Grades 10-12