Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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In question 23, they are asking me to explain why the function is discontinuous at the given number a and to sketch the graph of the function. I am not entirely sure how the function in problem 23 is discontinuous, any help as to how to solve the problem would be much appreciated.
![### Educational Content on Piecewise Functions and Discontinuities
#### Function Definitions
**21.**
\[ f(x) = 2^x \quad \text{if} \, x > -1 \]
**22.**
\[ f(x) = \begin{cases}
\frac{x^2 - x}{x^2 - 1} & \text{if} \, x \neq 1 \\
1 & \text{if} \, x = 1
\end{cases} \]
**23.**
\[ f(x) = \begin{cases}
\cos x & \text{if} \, x < 0 \\
0 & \text{if} \, x = 0 \\
1 - x^2 & \text{if} \, x > 0
\end{cases} \]
**24.**
\[ f(x) = \begin{cases}
\frac{2x^2 - 5x - 3}{x - 3} & \text{if} \, x \neq 3 \\
6 & \text{if} \, x = 3
\end{cases} \]
---
#### Problem Section
**25-26.**
(a) Show that \( f \) has a removable discontinuity at \( x = 3 \).
(b) Redefine \( f(3) \) so that \( f \) is continuous at \( x = 3 \) (and the discontinuity is “removed”).
**25.**
\[ f(x) = \frac{x - 3}{x^2 - 9} \]
**26.**
\[ f(x) = \frac{x^2 - 7x + 10}{x - 5} \]
---
### Explanation
- **Graphs and Diagrams**: There are no explicit graphs or diagrams in the image. The text involves piecewise-defined functions which may exhibit discontinuities at specified points.
- **Removable Discontinuities**: The exercise involves identifying and redefining functions to eliminate removable discontinuities at specific \( x \) values, suggesting exploration of limits and continuity in piecewise functions.
This educational content provides an exploration of mathematical concepts such as continuity, limits, and functional redefinition to create continuous functions from piecewise definitions.](https://content.bartleby.com/qna-images/question/c59ea548-1a11-4ac7-baf4-61812d799a32/3dfa51b2-e9c6-4823-89fc-dbf1819d026f/1wx6an_processed.jpeg)
Transcribed Image Text:### Educational Content on Piecewise Functions and Discontinuities
#### Function Definitions
**21.**
\[ f(x) = 2^x \quad \text{if} \, x > -1 \]
**22.**
\[ f(x) = \begin{cases}
\frac{x^2 - x}{x^2 - 1} & \text{if} \, x \neq 1 \\
1 & \text{if} \, x = 1
\end{cases} \]
**23.**
\[ f(x) = \begin{cases}
\cos x & \text{if} \, x < 0 \\
0 & \text{if} \, x = 0 \\
1 - x^2 & \text{if} \, x > 0
\end{cases} \]
**24.**
\[ f(x) = \begin{cases}
\frac{2x^2 - 5x - 3}{x - 3} & \text{if} \, x \neq 3 \\
6 & \text{if} \, x = 3
\end{cases} \]
---
#### Problem Section
**25-26.**
(a) Show that \( f \) has a removable discontinuity at \( x = 3 \).
(b) Redefine \( f(3) \) so that \( f \) is continuous at \( x = 3 \) (and the discontinuity is “removed”).
**25.**
\[ f(x) = \frac{x - 3}{x^2 - 9} \]
**26.**
\[ f(x) = \frac{x^2 - 7x + 10}{x - 5} \]
---
### Explanation
- **Graphs and Diagrams**: There are no explicit graphs or diagrams in the image. The text involves piecewise-defined functions which may exhibit discontinuities at specified points.
- **Removable Discontinuities**: The exercise involves identifying and redefining functions to eliminate removable discontinuities at specific \( x \) values, suggesting exploration of limits and continuity in piecewise functions.
This educational content provides an exploration of mathematical concepts such as continuity, limits, and functional redefinition to create continuous functions from piecewise definitions.
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