Consider the following. y = x - 3+ 2 3+²/² y X 5 5 (a) Use the graph to determine any x-intercepts of the graph of the rational function. (x, y) = (smaller x-value) (x, y) = (larger x-value) (b) Set y = and solve the resulting equation to confirm your result in part (a). (x, y) = (smaller x-value) (x, y) = (larger x-value)
Consider the following. y = x - 3+ 2 3+²/² y X 5 5 (a) Use the graph to determine any x-intercepts of the graph of the rational function. (x, y) = (smaller x-value) (x, y) = (larger x-value) (b) Set y = and solve the resulting equation to confirm your result in part (a). (x, y) = (smaller x-value) (x, y) = (larger x-value)
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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![### Rational Function Analysis and Graph Interpretation
#### Function and Graph
Consider the following rational function:
\[ y = x - 3 + \frac{2}{x} \]
Below is the graph of the function.
The graph shows an asymptotic behavior near the y-axis and horizontal asymptotes as \( x \) approaches positive and negative infinity. The function undergoes critical points and intercepts that help to define its shape and characteristics.
#### Analyzing the Graph
1. **Determine the x-intercepts of the graph:**
(a) Use the graph to determine any x-intercepts of the rational function.
- \((x, y) = \_\_\_\_\_\_\) (smaller x-value)
- \((x, y) = \_\_\_\_\_\_\) (larger x-value)
2. **Solve algebraically for confirmation:**
(b) Set \( y = 0 \) and solve the resulting equation to confirm your result in part (a).
\[ 0 = x - 3 + \frac{2}{x} \]
Solve for \( x \) to find the x-intercepts.
- \((x, y) = \_\_\_\_\_\_\) (smaller x-value)
- \((x, y) = \_\_\_\_\_\_\) (larger x-value)
#### Detailed Explanation of the Graph
- **Axes:** The graph includes both the x-axis and y-axis ranging from \(-5\) to \(5\).
- **Curve Behavior:** The curve crosses both the positive and negative sides of the axes, demonstrating the transitions and asymptotes.
By examining and solving the rational function, one can identify the x-intercepts and confirm them algebraically to ensure accuracy and comprehension of the function's behavior.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff2367e66-1ae4-4371-aa7b-db391deab840%2F02fafdb3-12cc-4975-b451-55bb2e71874e%2Fcn4xa84_processed.png&w=3840&q=75)
Transcribed Image Text:### Rational Function Analysis and Graph Interpretation
#### Function and Graph
Consider the following rational function:
\[ y = x - 3 + \frac{2}{x} \]
Below is the graph of the function.
The graph shows an asymptotic behavior near the y-axis and horizontal asymptotes as \( x \) approaches positive and negative infinity. The function undergoes critical points and intercepts that help to define its shape and characteristics.
#### Analyzing the Graph
1. **Determine the x-intercepts of the graph:**
(a) Use the graph to determine any x-intercepts of the rational function.
- \((x, y) = \_\_\_\_\_\_\) (smaller x-value)
- \((x, y) = \_\_\_\_\_\_\) (larger x-value)
2. **Solve algebraically for confirmation:**
(b) Set \( y = 0 \) and solve the resulting equation to confirm your result in part (a).
\[ 0 = x - 3 + \frac{2}{x} \]
Solve for \( x \) to find the x-intercepts.
- \((x, y) = \_\_\_\_\_\_\) (smaller x-value)
- \((x, y) = \_\_\_\_\_\_\) (larger x-value)
#### Detailed Explanation of the Graph
- **Axes:** The graph includes both the x-axis and y-axis ranging from \(-5\) to \(5\).
- **Curve Behavior:** The curve crosses both the positive and negative sides of the axes, demonstrating the transitions and asymptotes.
By examining and solving the rational function, one can identify the x-intercepts and confirm them algebraically to ensure accuracy and comprehension of the function's behavior.
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