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Chapter 4: Problem 274

Find the horizontal and vertical asymptotes. \(f(x)=\frac{x^{2}+3}{x^{2}+1}\)

Short Answer

Expert verified
Vertical asymptotes: None, Horizontal asymptote: \(y=1\).

Step by step solution

01

Identify Vertical Asymptotes

Vertical asymptotes occur where the denominator equals zero and the numerator isn't zero at the same point. Set the denominator of the function equal to zero and solve for the values of \(x\).\[x^2 + 1 = 0\]This gives us:\[x^2 = -1\]Since \(x^2 = -1\) has no real solutions, there are no vertical asymptotes.
02

Identify Horizontal Asymptotes

Horizontal asymptotes are determined by comparing the degrees of the numerator and the denominator. Both the numerator \(x^2 + 3\) and the denominator \(x^2 + 1\) are degree 2 polynomials.Since the degrees are the same, the horizontal asymptote is determined by the leading coefficients. The leading coefficient of the numerator is 1, and the leading coefficient of the denominator is 1.Thus, the horizontal asymptote is:\[y = \frac{1}{1} = 1\]

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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 straight lines that occur where a function's denominator is equal to zero. Imagine a graph that shoots straight up or plummets straight down as it approaches this line. A vertical asymptote represents points where the function is undefined. To find vertical asymptotes, you set the denominator of a rational function equal to zero, then solve for the variable.
  • If the equation has real solutions, those values are where the vertical asymptotes lie.
  • If the denominator equals zero and the numerator also equals zero at the same point, that often indicates a hole instead of a vertical asymptote.
  • In our example function, \(f(x)=\frac{x^{2}+3}{x^{2}+1}\), the denominator \(x^2 + 1\) equals zero, when \(x^2 = -1\), which doesn't have real solutions. Therefore, no vertical asymptotes exist for this function.
Horizontal Asymptotes
Horizontal asymptotes are horizontal lines that the graph of the function approaches as \(x\) becomes very large or very small. They demonstrate the function's end behavior. To determine the presence of a horizontal asymptote, you generally compare the degrees of the polynomials in the numerator and the denominator.
  • If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \(y = 0\).
  • If both degrees are equal, the horizontal asymptote is \(y = \frac{a}{b}\), where \(a\) and \(b\) are the leading coefficients of the numerator and denominator.
  • If the degree of the numerator is greater than that of the denominator, there is no horizontal asymptote.
  • In our example, both \(x^2 + 3\) and \(x^2 + 1\) are degree 2, with leading coefficients of 1. Thus, the asymptote is \(y = 1\).
Degree of Polynomials
The degree of a polynomial is the highest power of the variable in the polynomial. It plays a crucial role in determining the behavior of polynomials, especially when analyzing asymptotes.
  • The degree helps us determine the horizontal behavior of a rational function, i.e., horizontal asymptotes.
  • In rational functions, the degree of the numerator and denominator dictate if and where horizontal and even slant asymptotes might exist.
  • By comparing the degrees, we understand if a function levels off to a horizontal asymptote or if it diverges.
  • In the function \(f(x)=\frac{x^{2}+3}{x^{2}+1}\), both the numerator and the denominator are degree 2, which helps us find the horizontal asymptote at \(y = 1\).
Understanding the degrees of polynomials helps predict the shape and characteristics of graphs, making analysis and prediction easier.

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