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Chapter 3: Problem 93

Give an example of a rational function that has vertical asymptote \(x=3 .\) Now give an example of one that has vertical asymptote \(x=3\) and horizontal asymptote \(y=2 .\) Now give an example of a rational function with vertical asymptotes \(x=1\) and \(x=-1,\) horizontal asymptote \(y=0,\) and \(x\) -intercept 4

Short Answer

Expert verified
1. \(f(x) = \frac{1}{x-3}\); 2. \(g(x) = \frac{2x}{x-3}\); 3. \(h(x) = \frac{x-4}{(x-1)(x+1)}\).

Step by step solution

01

Understanding Vertical Asymptotes

A vertical asymptote occurs in a rational function where the function approaches infinity as the input approaches a certain value. This typically results from setting the denominator equal to zero. For a vertical asymptote at \(x = 3\), the denominator should have a factor of \((x - 3)\).
02

Construct a Function with Vertical Asymptote at \(x=3\)

A simple rational function with a vertical asymptote at \(x=3\) is \(f(x) = \frac{1}{x-3}\). Here, as \(x\) approaches 3, \(f(x)\) approaches infinity, giving a vertical asymptote at \(x=3\).
03

Adding a Horizontal Asymptote to the Function

To have a horizontal asymptote at \(y=2\), the degrees of the polynomial in the numerator and denominator should be equal, and the leading coefficients of these polynomials will define the horizontal asymptote. A function satisfying these conditions can be \(g(x) = \frac{2x}{x-3}\). As \(x\) approaches infinity, \(g(x)\) approaches 2.
04

Understanding Horizontal Asymptote

The rational function \(g(x) = \frac{2x}{x-3}\) has a horizontal asymptote \(y=2\) because the degrees of the numerator and denominator polynomials are the same, and the leading coefficient of the numerator is 2, resulting in the horizontal asymptote \(y=2\).
05

Construct a Function with Multiple Vertical Asymptotes

For vertical asymptotes at \(x=1\) and \(x=-1\), the denominator should contain the factors \((x-1)(x+1)\). A simple function meeting this requirement is \(h(x) = \frac{x}{(x-1)(x+1)}\). As \(x\) approaches 1 and -1, \(h(x)\) approaches infinity, confirming the vertical asymptotes.
06

Ensuring a Horizontal Asymptote at \(y=0\)

For a horizontal asymptote at \(y=0\), the degree of the polynomial in the denominator must be greater than the degree of the polynomial in the numerator. This condition is already satisfied in \(h(x) = \frac{x}{(x-1)(x+1)}\), where the degree of the denominator is 2 and the degree of the numerator is 1.
07

Adding an \(x\)-Intercept to the Function

To achieve an \(x\)-intercept at \(x=4\) for \(h(x) = \frac{x}{(x-1)(x+1)}\), the numerator must become zero at \(x=4\). The current numerator \(x\) needs to be modified to \((x-4)\). Thus, the function becomes \(h(x) = \frac{x-4}{(x-1)(x+1)}\), satisfying all problem conditions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vertical Asymptotes
Vertical asymptotes are lines where a rational function's value becomes infinitely large or small. This typically happens when the denominator of the function equals zero, as a zero in the denominator makes the function undefined. For instance, in the rational function \( f(x) = \frac{1}{x-3} \), the denominator becomes zero when \( x = 3 \). Therefore, the function approaches infinity as \( x \) gets closer to 3, creating a vertical asymptote at \( x = 3 \).Here are a few things to remember about vertical asymptotes:
  • They occur at points where the denominator is zero, but not the numerator.
  • They indicate where the function is undefined and unbounded.
  • The function will never actually touch or cross a vertical asymptote.
Knowing how to identify and work with vertical asymptotes is crucial when analyzing the behavior of rational functions.
Horizontal Asymptotes
Horizontal asymptotes represent the behavior of a rational function as \( x \) approaches infinity or negative infinity. These are horizontal lines that the function approaches, but might not necessarily touch, in extreme x-values. To determine the horizontal asymptote, you need to look at the degrees of the polynomials in the numerator and the denominator.
  • If the degree of the numerator and denominator is the same, the horizontal asymptote is \( y = \frac{a}{b} \), where \( a \) and \( b \) are the leading coefficients.
  • If the degree of the numerator is less than that of the denominator, the horizontal asymptote is \( y = 0 \).
  • If the numerator's degree is greater, there is no horizontal asymptote (instead, check for an oblique asymptote).
For example, in the function \( g(x) = \frac{2x}{x-3} \), both the numerator and denominator are of degree 1. Hence, the horizontal asymptote is at \( y = 2 \). Horizontal asymptotes help us understand the limits of a function's behavior.
x-Intercepts
The x-intercepts of a rational function are the points where the function crosses the x-axis. This occurs when the numerator equals zero while the denominator does not. For a function \( h(x) = \frac{x-4}{(x-1)(x+1)} \), the x-intercept is found by setting the numerator equal to zero: \( x-4 = 0 \). Solving this equation, we get \( x = 4 \), indicating that the function crosses the x-axis at this point.Here are some vital points about x-intercepts to keep in mind:
  • They are found by setting the numerator of the rational function equal to zero.
  • Only valid when the entire function does not become undefined (denominator cannot be zero at these points).
  • Graphically, they are where the function touches or crosses the x-axis on a graph.
Understanding x-intercepts is essential for accurately graphing rational functions and solving equations.
Polynomial Degrees
The degree of a polynomial function is the highest power of the variable \( x \) that appears in the polynomial. In rational functions, comparing the degrees of the numerator and denominator polynomials is key in determining the behavior of the function, particularly its asymptotic behavior.
  • The degree of a polynomial is significant when analyzing asymptotes.
  • It鈥檚 important to understand and compare these degrees: for horizontal asymptotes, and for understanding end-behavior.
  • Degrees affect the intercepts and possibly the holes in the graph.
For example, in the function \( g(x) = \frac{2x}{x-3} \), the degree of both the numerator and the denominator is 1, which helps us find that the horizontal asymptote is at \( y=2 \). Comfort with polynomial degrees is vital for mastering rational function behaviors.

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Most popular questions from this chapter

Find the intercepts and asymptotes, and then sketch a graph of the rational function and state the domain and range. Use a graphing device to confirm your answer. $$r(x)=\frac{3 x^{2}+6}{x^{2}-2 x-3}$$

Graphing Quadratic Functions A quadratic function \(f\) is given. (a) Express \(f\) in standard form. (b) Find the vertex and \(x\) and \(y\) -intercepts of \(f .\) (c) Sketch a graph of \(f .\) (d) Find the domain and range of \(f\). $$f(x)=-x^{2}-4 x+4$$

Graph the rational function, and find all vertical asymptotes, \(x\) - and \(y\) -intercepts, and local extrema, correct to the nearest tenth. Then use long division to find a polynomial that has the same end behavior as the rational function, and graph both functions in a sufficiently large viewing rectangle to verify that the end behaviors of the polynomial and the rational function are the same. $$y=\frac{2 x^{2}-5 x}{2 x+3}$$

Maximum and Minimum Values A quadratic function \(f\) is given. (a) Express \(f\) in standard form. (b) Sketch a graph of \(f .\) (c) Find the maximum or minimum value of \(f .\) $$f(x)=x^{2}+2 x-1$$

The effectiveness of a television commercial depends on how many times a viewer watches it. After some experiments an advertising agency found that if the effectiveness \(E\) is measured on a scale of 0 to \(10,\) then $$E(n)=\frac{2}{3} n-\frac{1}{90} n^{2}$$ where \(n\) is the number of times a viewer watches a given commercial. For a commercial to have maximum effectiveness, how many times should a viewer watch it?
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