https://file.aiquickdraw.com/mathbot/file/1726975107913onnlwdps.jpg

The image shows a mathematical problem involving the identification of a jump discontinuity in a piecewise function \( f(x) \) at \( x = 5 \). Here's a step-by-step approach to address this.

### Problem Breakdown:

1. **Definition of a Jump Discontinuity:**
   A function \( f(x) \) has a jump discontinuity at \( x = a \) if:
   - \( \lim_{x \to a^-} f(x) \) exists,
   - \( \lim_{x \to a^+} f(x) \) exists, and
   - The left and right limits are **not equal**.

2. **Piecewise Function Definition:**
   \[
   f(x) =
   \begin{cases}
   x^2 + 3x + 7 & \text{if } x < 5, \\
   7 & \text{if } x = 5, \\
   -2x + 4 & \text{if } x > 5.
   \end{cases}
   \]

3. **Task:** 
   We need to show that \( f(x) \) has a jump discontinuity at \( x = 5 \) by calculating the left-hand limit \( \lim_{x \to 5^-} f(x) \) and the right-hand limit \( \lim_{x \to 5^+} f(x) \).

### Step-by-Step Solution:

#### 1. **Left-hand Limit** (\( x \to 5^- \)):
For \( x < 5 \), \( f(x) = x^2 + 3x + 7 \).
\[
\lim_{x \to 5^-} f(x) = \lim_{x \to 5^-} (x^2 + 3x + 7).
\]
Substitute \( x = 5 \) into the expression:
\[
f(5^-) = 5^2 + 3(5) + 7 = 25 + 15 + 7 = 47.
\]

#### 2. **Right-hand Limit** (\( x \to 5^+ \)):
For \( x > 5 \), \( f(x) = -2x + 4 \).
\[
\lim_{x \to 5^+} f(x) = \lim_{x \to 5^+} (-2x + 4).
\]
Substitute \( x = 5 \) into the expression:
\[
f(5^+) = -2(5) + 4 = -10 + 4 = -6.
\]

#### 3. **Check for Jump Discontinuity:**
Since \( \lim_{x \to 5^-} f(x) = 47 \) and \( \lim_{x \to 5^+} f(x) = -6 \), and these two limits are **not equal**, there is a jump discontinuity at \( x = 5 \).

### Graphing \( f(x) \):
The graph would show different behaviors for \( x < 5 \), \( x = 5 \), and \( x > 5 \):
- For \( x < 5 \), the graph follows the parabola \( x^2 + 3x + 7 \).
- At \( x = 5 \), the function takes a constant value \( 7 \).
- For \( x > 5 \), the graph follows the line \( -2x + 4 \).

### Would you like more details or any specific explanations on this problem?

Here are 5 related questions you can explore:

1. What are the conditions for a function to have a removable discontinuity?
2. How do you classify other types of discontinuities (e.g., infinite discontinuity)?
3. What is the difference between a jump discontinuity and a removable discontinuity?
4. How can you extend this piecewise function to make it continuous at \( x = 5 \)?
5. What is the significance of limits when analyzing the behavior of piecewise functions?

**Tip:** When evaluating limits for piecewise functions, always make sure to consider the specific domain for each piece of the function carefully.

Show that f(x) has a jump discontinuity at x = 5 by calculating the limits from the left and right at x = 5. The piecewise function f(x) is defined as follows:
- For x < 5: f(x) = x^2 + 3x + 7
- For x = 5: f(x) = 7
- For x > 5: f(x) = -2x + 4

Learn how to show that the function f(x) has a jump discontinuity at x = 5 by calculating left-hand and right-hand limits. This piecewise function problem is ideal for students learning about limits and discontinuities.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation