Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
FInd Derivative using the definition of derivatives
![### Function Description
The image shows a mathematical function defined as follows:
\[ f(x) = \frac{5 + x}{3 - x} \]
This function, \( f(x) \), is a rational function where the numerator is \( 5 + x \) and the denominator is \( 3 - x \). The function is undefined when the denominator equals zero, so solving \( 3 - x = 0 \) gives \( x = 3 \) as a point of discontinuity.
### Graphical Explanation
If this were graphed, here’s what you would expect:
- **Vertical Asymptote:** At \( x = 3 \), the function has a vertical asymptote due to the division by zero in the denominator.
- **Horizontal Asymptote:** As \( x \) approaches infinity or negative infinity, the behavior of this rational function can be analyzed by dividing the leading terms in the numerator and denominator, resulting in a horizontal asymptote at \( y = 1 \).
- **Intercepts:**
- **Y-intercept:** To find the y-intercept, set \( x = 0 \), which gives \( f(0) = \frac{5 + 0}{3 - 0} = \frac{5}{3} \).
- **X-intercept:** To find the x-intercept, set \( f(x) = 0 \), which occurs when the numerator is zero: \( 5 + x = 0 \) leading to \( x = -5 \).
This function's graph demonstrates typical behavior for rational functions, with specific interest at points of intersection with axes and asymptotic behavior near lines where it becomes undefined or increasingly closer yet never reaching those values.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faf8dd5bb-1368-4a10-9929-9294802d0b74%2Fdef91e7c-13ec-4507-8211-7f4ac160538f%2Fjl5jtj_processed.png&w=3840&q=75)
Transcribed Image Text:### Function Description
The image shows a mathematical function defined as follows:
\[ f(x) = \frac{5 + x}{3 - x} \]
This function, \( f(x) \), is a rational function where the numerator is \( 5 + x \) and the denominator is \( 3 - x \). The function is undefined when the denominator equals zero, so solving \( 3 - x = 0 \) gives \( x = 3 \) as a point of discontinuity.
### Graphical Explanation
If this were graphed, here’s what you would expect:
- **Vertical Asymptote:** At \( x = 3 \), the function has a vertical asymptote due to the division by zero in the denominator.
- **Horizontal Asymptote:** As \( x \) approaches infinity or negative infinity, the behavior of this rational function can be analyzed by dividing the leading terms in the numerator and denominator, resulting in a horizontal asymptote at \( y = 1 \).
- **Intercepts:**
- **Y-intercept:** To find the y-intercept, set \( x = 0 \), which gives \( f(0) = \frac{5 + 0}{3 - 0} = \frac{5}{3} \).
- **X-intercept:** To find the x-intercept, set \( f(x) = 0 \), which occurs when the numerator is zero: \( 5 + x = 0 \) leading to \( x = -5 \).
This function's graph demonstrates typical behavior for rational functions, with specific interest at points of intersection with axes and asymptotic behavior near lines where it becomes undefined or increasingly closer yet never reaching those values.
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