Given: $f(x)=\frac{3x-4}{x+1}$ Express Your Answers For The Following:a. Domain In Interval Notation? B. Vertical Asymptote(s)? C. Horizontal/Slant Asymptote? D. $x$-intercept(s)? E. $y$-intercept? F. Graph The Function

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Introduction

In this article, we will delve into the analysis of a given rational function, f(x)=3x−4x+1f(x)=\frac{3x-4}{x+1}. We will explore the domain, vertical and horizontal/slant asymptotes, xx-intercepts, yy-intercept, and graph the function. This analysis will provide a comprehensive understanding of the function's behavior and characteristics.

Domain in Interval Notation

The domain of a rational function consists of all real numbers except those that make the denominator equal to zero. To find the domain, we need to determine the values of xx that make the denominator, x+1x+1, equal to zero.

  • Step 1: Set the denominator equal to zero: x+1=0x+1=0
  • Step 2: Solve for xx: x=−1x=-1

The domain of the function is all real numbers except x=−1x=-1. In interval notation, the domain is represented as:

(−∞,−1)∪(−1,∞)(-\infty, -1) \cup (-1, \infty)

Vertical Asymptote(s)

A vertical asymptote occurs when the denominator of the rational function is equal to zero. In this case, the denominator is x+1x+1, which equals zero when x=−1x=-1. Therefore, the vertical asymptote is:

x=−1x=-1

Horizontal/Slant Asymptote

To determine the horizontal or slant asymptote, we need to compare the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y=0y=0. If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is the ratio of the leading coefficients. If the degree of the numerator is greater than the degree of the denominator, the slant asymptote is a polynomial of degree n−1n-1, where nn is the degree of the numerator.

In this case, the degree of the numerator is 1, and the degree of the denominator is 1. The leading coefficients are 3 and 1, respectively. Therefore, the horizontal asymptote is:

y=31=3y=\frac{3}{1}=3

x-Intercept(s)

The xx-intercept occurs when the function is equal to zero. To find the xx-intercept, we need to set the function equal to zero and solve for xx.

  • Step 1: Set the function equal to zero: 3x−4x+1=0\frac{3x-4}{x+1}=0
  • Step 2: Solve for xx: 3x−4=0⇒x=433x-4=0 \Rightarrow x=\frac{4}{3}

The xx-intercept is:

x=43x=\frac{4}{3}

y-Intercept

The yy-intercept occurs when x=0x=0. To find the yy-intercept, we need to substitute x=0x=0 into the function.

  • Step 1: Substitute x=0x=0 into the function: f(0)=3(0)−40+1=−4f(0)=\frac{3(0)-4}{0+1}=-4

The yy-intercept is:

y=−4y=-4

Graphing the Function

To graph the function, we can use the information obtained from the analysis. The graph will have a vertical asymptote at x=−1x=-1, a horizontal asymptote at y=3y=3, an xx-intercept at x=43x=\frac{4}{3}, and a yy-intercept at y=−4y=-4.

Here is a graph of the function:

### Graph of the Function

The graph of the function $f(x)=\frac{3x-4}{x+1}$ has a vertical asymptote at $x=-1$, a horizontal asymptote at $y=3$, an $x$-intercept at $x=\frac{4}{3}$, and a $y$-intercept at $y=-4$.

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**Q&A: Analyzing the Rational Function**
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**Q: What is the domain of the rational function $f(x)=\frac{3x-4}{x+1}$?**
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A: The domain of the rational function is all real numbers except $x=-1$. In interval notation, the domain is represented as:

$(-\infty, -1) \cup (-1, \infty)$

**Q: What is the vertical asymptote of the rational function?**
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A: The vertical asymptote of the rational function is $x=-1$.

**Q: What is the horizontal/slant asymptote of the rational function?**
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A: The horizontal asymptote of the rational function is $y=3$.

**Q: What is the x-intercept of the rational function?**
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A: The $x$-intercept of the rational function is $x=\frac{4}{3}$.

**Q: What is the y-intercept of the rational function?**
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A: The $y$-intercept of the rational function is $y=-4$.

**Q: How do I graph the rational function?**
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A: To graph the rational function, you can use the information obtained from the analysis. The graph will have a vertical asymptote at $x=-1$, a horizontal asymptote at $y=3$, an $x$-intercept at $x=\frac{4}{3}$, and a $y$-intercept at $y=-4$.

**Q: What are some common mistakes to avoid when analyzing rational functions?**
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A: Some common mistakes to avoid when analyzing rational functions include:

*   Not considering the domain of the function
*   Not identifying the vertical asymptote
*   Not comparing the degrees of the numerator and denominator
*   Not finding the horizontal or slant asymptote
*   Not finding the $x$-intercept and $y$-intercept

**Q: How do I determine the degree of the numerator and denominator of a rational function?**
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A: To determine the degree of the numerator and denominator of a rational function, you can count the number of terms in the numerator and denominator. The degree of the numerator is the highest power of $x$ in the numerator, and the degree of the denominator is the highest power of $x$ in the denominator.

**Q: What is the significance of the horizontal or slant asymptote of a rational function?**
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A: The horizontal or slant asymptote of a rational function represents the behavior of the function as $x$ approaches infinity or negative infinity. It provides a way to understand the long-term behavior of the function.

**Q: How do I find the x-intercept and y-intercept of a rational function?**
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A: To find the $x$-intercept and $y$-intercept of a rational function, you can set the function equal to zero and solve for $x$, or substitute $x=0$ into the function, respectively.

**Q: What are some real-world applications of rational functions?**
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A: Rational functions have many real-world applications, including:

*   Modeling population growth and decline
*   Analyzing financial data
*   Studying the behavior of electrical circuits
*   Modeling the motion of objects
*   Analyzing the behavior of chemical reactions

By understanding the properties and behavior of rational functions, you can apply this knowledge to a wide range of real-world problems and applications.