6. Which of the following functions f has a removable discontinuity at a? If the discontinuity is removable, find a function g that agrees with f for x ≠ a and is continuous at a. (If an answer does not exist, enter DNE.) (a) f(x) = x^4 − 1 / x − 1 , a = 1 The discontinuity is removable. or The discontinuity is not removable. ? g(x) = (b) f(x) = x^3 − x^2 − 12x / x − 4 , a = 4 The discontinuity is removable. or The discontinuity is not removable. ? g(x) = (c) f(x) = sin(x) , a = 𝜋 (Recall that h(x) means the largest integer that is less than or equal to h(x).) The discontinuity is removable. or The discontinuity is not removable. ? g(x) =

Let's analyze each function to determine if the discontinuity at the specified \( a \) is removable or not. A discontinuity is removable if we can redefine the function at \( a \) such that the function becomes continuous at that point.

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### (a) \( f(x) = \frac{x^4 - 1}{x - 1} \), \( a = 1 \)

We need to check if there is a removable discontinuity at \( x = 1 \).

- **Step 1**: Factor the numerator.  
  \( x^4 - 1 \) is a difference of squares:
  \[
  x^4 - 1 = (x^2 - 1)(x^2 + 1) = (x - 1)(x + 1)(x^2 + 1)
  \]

- **Step 2**: Simplify the function.  
  Now, rewrite \( f(x) \):
  \[
  f(x) = \frac{(x - 1)(x + 1)(x^2 + 1)}{x - 1}
  \]
  For \( x \neq 1 \), we can cancel out \( (x - 1) \):
  \[
  f(x) = (x + 1)(x^2 + 1)
  \]

- **Step 3**: Define \( g(x) \), which agrees with \( f(x) \) for \( x \neq 1 \) and is continuous at \( x = 1 \).  
  Since the function simplifies to \( g(x) = (x + 1)(x^2 + 1) \), we can evaluate \( g(1) \):
  \[
  g(1) = (1 + 1)(1^2 + 1) = 2 \times 2 = 4
  \]
  Therefore, the discontinuity is **removable**, and the function \( g(x) \) is:
  \[
  g(x) = (x + 1)(x^2 + 1)
  \]

---

### (b) \( f(x) = \frac{x^3 - x^2 - 12x}{x - 4} \), \( a = 4 \)

We need to check if there is a removable discontinuity at \( x = 4 \).

- **Step 1**: Factor the numerator.  
  \( x^3 - x^2 - 12x \) can be factored by first factoring out \( x \):
  \[
  x^3 - x^2 - 12x = x(x^2 - x - 12)
  \]
  Now factor \( x^2 - x - 12 \):
  \[
  x^2 - x - 12 = (x - 4)(x + 3)
  \]
  So, we have:
  \[
  f(x) = \frac{x(x - 4)(x + 3)}{x - 4}
  \]

- **Step 2**: Simplify the function.  
  For \( x \neq 4 \), we can cancel out \( (x - 4) \):
  \[
  f(x) = x(x + 3)
  \]

- **Step 3**: Define \( g(x) \), which agrees with \( f(x) \) for \( x \neq 4 \) and is continuous at \( x = 4 \).  
  Since the function simplifies to \( g(x) = x(x + 3) \), we can evaluate \( g(4) \):
  \[
  g(4) = 4(4 + 3) = 4 \times 7 = 28
  \]
  Therefore, the discontinuity is **removable**, and the function \( g(x) \) is:
  \[
  g(x) = x(x + 3)
  \]

---

### (c) \( f(x) = \sin(x) \), \( a = \pi \)

- **Step 1**: Analyze the function.  
  The function \( f(x) = \sin(x) \) is continuous everywhere in its domain, including at \( x = \pi \). There is no discontinuity at \( x = \pi \).

Thus, the discontinuity is **not removable**, and the function \( g(x) \) does not need to be redefined since there is no discontinuity.

---

### Final Answers:

- **(a)** The discontinuity is **removable**.  
  \( g(x) = (x + 1)(x^2 + 1) \)
  
- **(b)** The discontinuity is **removable**.  
  \( g(x) = x(x + 3) \)

- **(c)** The discontinuity is **not removable** (since there is no discontinuity).  
  \( g(x) = \text{DNE} \)

---

Do you have any questions or would you like further details? 

Here are 5 related questions to deepen understanding:
1. How do you factor a cubic polynomial to find removable discontinuities?
2. What is the difference between removable and non-removable discontinuities?
3. How do you check if a function is continuous at a point?
4. Can all rational functions have removable discontinuities?
5. What role do limits play in determining removable discontinuities?

**Tip**: Always check if you can simplify a function by factoring when looking for removable discontinuities.

6. Which of the following functions f has a removable discontinuity at a? If the discontinuity is removable, find a function g that agrees with f for x ≠ a and is continuous at a. (a) f(x) = x^4 − 1 / x − 1 , a = 1 The discontinuity is removable. or The discontinuity is not removable. ? g(x) = (b) f(x) = x^3 − x^2 − 12x / x − 4 , a = 4 The discontinuity is removable. or The discontinuity is not removable. ? g(x) = (c) f(x) = sin(x) , a = 𝜋 (Recall that h(x) means the largest integer that is less than or equal to h(x).) The discontinuity is removable. or The discontinuity is not removable. ? g(x) =

Explore how to identify removable discontinuities in rational functions with examples such as f(x) = x^4 − 1 / x − 1 and f(x) = x^3 − x^2 − 12x / x − 4. Learn how to simplify functions, apply limits, and understand continuity at points of discontinuity. Suitable for high school math students in Grades 11-12.

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