Let $f(x)=\frac{x^2-x-2}{x-1}$.Find The Domain, Vertical Asymptotes, Slant Asymptote, And Select The Correct Graph.- Domain: $\square$ (intervals)- Vertical Asymptotes: $\square$- Slant Asymptote: $\square$Which Of
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Introduction
The given rational function is $f(x)=\frac{x^2-x-2}{x-1}$. To understand the behavior of this function, we need to find its domain, vertical asymptotes, slant asymptote, and select the correct graph. In this article, we will delve into each of these aspects and provide a detailed explanation.
Domain
The domain of a function is the set of all possible input values for which the function is defined. In the case of a rational function, the domain is all real numbers except for the values that make the denominator zero. To find the domain of $f(x)=\frac{x^2-x-2}{x-1}$, we need to determine the values of x that make the denominator zero.
Finding the Values that Make the Denominator Zero
To find the values that make the denominator zero, we set the denominator equal to zero and solve for x.
Solving for x, we get:
Therefore, the value that makes the denominator zero is x = 1.
Domain of the Function
Since the denominator is zero at x = 1, the function is undefined at this point. Therefore, the domain of the function is all real numbers except for x = 1.
Vertical Asymptotes
Vertical asymptotes are the vertical lines that the graph of a function approaches but never touches. In the case of a rational function, vertical asymptotes occur at the values of x that make the denominator zero.
Finding the Vertical Asymptotes
We have already found that the value that makes the denominator zero is x = 1. Therefore, the vertical asymptote is x = 1.
Slant Asymptote
A slant asymptote is a slant line that the graph of a function approaches but never touches. In the case of a rational function, a slant asymptote occurs when the degree of the numerator is exactly one more than the degree of the denominator.
Finding the Slant Asymptote
To find the slant asymptote, we need to divide the numerator by the denominator using long division or synthetic division.
From the result of the division, we can see that the slant asymptote is y = x + 2.
Selecting the Correct Graph
Now that we have found the domain, vertical asymptotes, and slant asymptote, we can select the correct graph.
Graph 1
Graph 1 has a vertical asymptote at x = 1 and a slant asymptote at y = x + 2.
Graph 2
Graph 2 has a vertical asymptote at x = 1 but no slant asymptote.
Graph 3
Graph 3 has no vertical asymptote at x = 1 and a slant asymptote at y = x + 2.
Graph 4
Graph 4 has no vertical asymptote at x = 1 and no slant asymptote.
Based on our analysis, the correct graph is Graph 1.
Conclusion
In this article, we found the domain, vertical asymptotes, slant asymptote, and selected the correct graph for the given rational function $f(x)=\frac{x^2-x-2}{x-1}$. We determined that the domain is all real numbers except for x = 1, the vertical asymptote is x = 1, the slant asymptote is y = x + 2, and the correct graph is Graph 1.
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Introduction
In our previous article, we explored the domain, vertical asymptotes, slant asymptote, and selected the correct graph for the given rational function $f(x)=\frac{x^2-x-2}{x-1}$. In this Q&A article, we will address some common questions and provide additional insights into the behavior of this function.
Q: What is the domain of the function?
A: The domain of the function is all real numbers except for x = 1, where the denominator is zero.
Q: Why is x = 1 not included in the domain?
A: The value x = 1 makes the denominator zero, which means the function is undefined at this point. Therefore, x = 1 is not included in the domain.
Q: What is the vertical asymptote of the function?
A: The vertical asymptote of the function is x = 1, where the denominator is zero.
Q: Why is there a vertical asymptote at x = 1?
A: The vertical asymptote occurs because the denominator is zero at x = 1, causing the function to approach infinity as x approaches 1 from either side.
Q: What is the slant asymptote of the function?
A: The slant asymptote of the function is y = x + 2, which is obtained by dividing the numerator by the denominator using long division or synthetic division.
Q: Why is there a slant asymptote at y = x + 2?
A: The slant asymptote occurs because the degree of the numerator is exactly one more than the degree of the denominator, causing the function to approach a linear asymptote as x approaches infinity.
Q: How do I select the correct graph?
A: To select the correct graph, look for the graph that has a vertical asymptote at x = 1 and a slant asymptote at y = x + 2.
Q: What if I'm still unsure about the correct graph?
A: If you're still unsure, try plotting the function using a graphing calculator or software to visualize the behavior of the function.
Q: Can I use this function in real-world applications?
A: Yes, this function can be used in real-world applications such as modeling population growth, chemical reactions, or electrical circuits.
Q: How do I extend this function to more complex rational functions?
A: To extend this function to more complex rational functions, you can use techniques such as factoring, canceling, or using long division or synthetic division to simplify the function.
Q: What are some common mistakes to avoid when working with rational functions?
A: Some common mistakes to avoid when working with rational functions include:
- Not checking for domain restrictions
- Not simplifying the function before graphing
- Not using the correct notation for the function
- Not considering the behavior of the function as x approaches infinity or negative infinity
Conclusion
In this Q&A article, we addressed some common questions and provided additional insights into the behavior of the given rational function $f(x)=\frac{x^2-x-2}{x-1}$. We hope this article has been helpful in clarifying any doubts and providing a deeper understanding of rational functions.