https://file.aiquickdraw.com/mathbot/file/17260995638114y7xo7rd.jpg

Let's solve the problem based on the function \( f(x) \) given in the image. The function is defined piecewise as:

\[
f(x) = 
\begin{cases} 
-x + b, & \text{if } x < 3 \\
1, & \text{if } x = 3 \\
\frac{-2}{x - b} - 1, & \text{if } x > 3 \text{ and } x \neq b
\end{cases}
\]

We need to address the following questions about the continuity and discontinuities of \( f(x) \) at \( x = 3 \) based on the parameter \( b \).

### Part (a): Continuity at \( x = 3 \)

For \( f(x) \) to be continuous at \( x = 3 \), the following must hold:
1. \( \lim_{x \to 3^-} f(x) = f(3) \)
2. \( \lim_{x \to 3^+} f(x) = f(3) \)
3. The limit values from the left and right must equal \( f(3) \).

**Left-hand limit:**
\[
\lim_{x \to 3^-} f(x) = \lim_{x \to 3^-} (-x + b) = -3 + b
\]

**Right-hand limit:**
\[
\lim_{x \to 3^+} f(x) = \lim_{x \to 3^+} \left( \frac{-2}{x - b} - 1 \right) = \frac{-2}{3 - b} - 1
\]

For continuity at \( x = 3 \), both the left and right limits must equal \( f(3) = 1 \).

So:
\[
-3 + b = 1 \quad \text{(left-hand limit)}
\]
\[
\frac{-2}{3 - b} - 1 = 1 \quad \text{(right-hand limit)}
\]

Solving the left-hand limit equation:
\[
b = 4
\]

For the right-hand limit:
\[
\frac{-2}{3 - b} - 1 = 1 \implies \frac{-2}{3 - b} = 2 \implies 3 - b = -1 \implies b = 4
\]

Thus, for \( f(x) \) to be continuous at \( x = 3 \), \( b = 4 \).

### Part (b): Removable discontinuity at \( x = 3 \)

A removable discontinuity occurs if the limits from the left and right exist and are equal but differ from \( f(3) \).

From part (a), we know that the limits are:
- \( \lim_{x \to 3^-} f(x) = -3 + b \)
- \( \lim_{x \to 3^+} f(x) = \frac{-2}{3 - b} - 1 \)

For the limits to be equal:
\[
-3 + b = \frac{-2}{3 - b} - 1
\]
Solving this equation will give us the value of \( b \) where there is a removable discontinuity. We already know that at \( b = 4 \), the function is continuous. Solving this equation for \( b \neq 4 \) gives us the removable discontinuity.

### Part (c): Infinite discontinuity at \( x = 3 \)

An infinite discontinuity would occur if the right-hand limit approaches \( \infty \) or \( -\infty \), which happens if the denominator of the expression \( \frac{-2}{3 - b} \) approaches zero.

This occurs when:
\[
3 - b = 0 \implies b = 3
\]
Thus, there is an infinite discontinuity at \( x = 3 \) when \( b = 3 \).

### Part (d): Jump discontinuity at \( x = 3 \)

A jump discontinuity occurs when the left-hand and right-hand limits exist but are not equal. From the previous results, we know:
- Left-hand limit: \( -3 + b \)
- Right-hand limit: \( \frac{-2}{3 - b} - 1 \)

For a jump discontinuity, these limits must exist but not be equal. This occurs for all \( b \) except \( b = 3 \) (infinite discontinuity) and \( b = 4 \) (continuous). Hence, the values of \( b \) where there is a jump discontinuity are all values of \( b \in (-\infty, 3) \cup (3, 4) \cup (4, \infty) \).

---

Would you like further details on these steps or explanations?

Here are 5 related questions to expand on this:
1. How can we determine if a function has a removable discontinuity at a different point?
2. What is the difference between a jump discontinuity and an infinite discontinuity?
3. How does the definition of a piecewise function impact its continuity analysis?
4. Can a function have both a removable and a jump discontinuity at the same point?
5. What are the graphical interpretations of different types of discontinuities?

**Tip**: When analyzing piecewise functions for continuity, always check both left-hand and right-hand limits carefully for any potential differences at the point in question.

Let f(x) be a piecewise function defined as follows:

f(x) = 
  -x + b, if x < 3
  1, if x = 3
  -2/(x - b) - 1, if x > 3 (and x ≠ b)

Find the values of b for which the function has specific types of continuity or discontinuity at x = 3.

Explore the continuity and types of discontinuities for the piecewise function f(x) at x = 3. Understand how limits and conditions define the function's behavior based on the parameter b. Suitable for high school math students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation