Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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please use the function to determined what is asked
![The given image contains the mathematical function \( f(x) = \frac{x^2 + x - 42}{x^2 + 5x - 14} \) and a list of tasks related to analyzing this function. Below is a transcription and explanation that could be used on an educational website:
---
## Analyzing the Rational Function \( f(x) \)
Consider the rational function:
\[ f(x) = \frac{x^2 + x - 42}{x^2 + 5x - 14} \]
To fully understand the behavior of this function, we need to analyze several key aspects:
**a. Vertical Asymptote(s)**:
Vertical asymptotes occur where the denominator of the function equals zero, as long as the numerator does not equal zero at the same points.
**b. Hole(s)**:
Holes in the graph occur where both the numerator and denominator are equal to zero at the same points. These are points cancelled out during simplification.
**c. Horizontal Asymptote(s)**:
To find horizontal asymptotes, we compare the degrees of the polynomials in the numerator and the denominator.
**d. Domain and Range**:
The domain of the function consists of all values of \( x \) for which the function is defined, excluding values that make the denominator zero. The range will be determined based on the output values of \( f(x) \).
**e. End Behavior**:
This describes what happens to the function values as \( x \) approaches \( \infty \) or \( -\infty \).
**f. Zero(s)**:
Zeros of the function occur where the numerator is zero but the denominator is not.
**g. Positive and Negative Intervals**:
These intervals explain where the function takes positive or negative values.
**h. Maximum(s) and Minimum(s)**:
Identify local and global maxima and minima, which reveal the peaks and troughs of the function.
**i. Intervals of Increase and Decrease**:
Determine where the function is increasing or decreasing, indicated by the derivative \( f'(x) \).
**j. Symmetry**:
Check for any symmetry by examining if the function is even, odd, or neither.
---
Make sure to follow these steps to analyze \( f(x) \) comprehensively, covering all the essential aspects listed above.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9f34e597-e324-4f60-b87e-69201cdf9380%2Fe907fb79-223e-4098-bb87-42660c472907%2Fyxv0i2r_processed.png&w=3840&q=75)
Transcribed Image Text:The given image contains the mathematical function \( f(x) = \frac{x^2 + x - 42}{x^2 + 5x - 14} \) and a list of tasks related to analyzing this function. Below is a transcription and explanation that could be used on an educational website:
---
## Analyzing the Rational Function \( f(x) \)
Consider the rational function:
\[ f(x) = \frac{x^2 + x - 42}{x^2 + 5x - 14} \]
To fully understand the behavior of this function, we need to analyze several key aspects:
**a. Vertical Asymptote(s)**:
Vertical asymptotes occur where the denominator of the function equals zero, as long as the numerator does not equal zero at the same points.
**b. Hole(s)**:
Holes in the graph occur where both the numerator and denominator are equal to zero at the same points. These are points cancelled out during simplification.
**c. Horizontal Asymptote(s)**:
To find horizontal asymptotes, we compare the degrees of the polynomials in the numerator and the denominator.
**d. Domain and Range**:
The domain of the function consists of all values of \( x \) for which the function is defined, excluding values that make the denominator zero. The range will be determined based on the output values of \( f(x) \).
**e. End Behavior**:
This describes what happens to the function values as \( x \) approaches \( \infty \) or \( -\infty \).
**f. Zero(s)**:
Zeros of the function occur where the numerator is zero but the denominator is not.
**g. Positive and Negative Intervals**:
These intervals explain where the function takes positive or negative values.
**h. Maximum(s) and Minimum(s)**:
Identify local and global maxima and minima, which reveal the peaks and troughs of the function.
**i. Intervals of Increase and Decrease**:
Determine where the function is increasing or decreasing, indicated by the derivative \( f'(x) \).
**j. Symmetry**:
Check for any symmetry by examining if the function is even, odd, or neither.
---
Make sure to follow these steps to analyze \( f(x) \) comprehensively, covering all the essential aspects listed above.
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