Problem 63 - Algebra for College Students

Question

In each case find a rational finction whose graph has the required asymptotes. Answers may vary. The graph has the \(x\) -axis as a horizontal asymptote and the y-axis as a vertical asymptote.

Step-by-Step Solution

Answer

A rational function with the given asymptotes is \(\frac{1}{x}\).

Step 1: Identify the Asymptotes

A rational function typically has the form \(\frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials. For this problem, we need the x-axis (\(y = 0\)) as a horizontal asymptote and the y-axis (\(x = 0\)) as a vertical asymptote.

Step 2: Determine the Conditions

For a horizontal asymptote at \(y = 0\), the degree of the polynomial in the numerator must be less than the degree of the polynomial in the denominator. To have the y-axis as a vertical asymptote, \(Q(x)\) must have a factor of \(x\).

Step 3: Form the Rational Function

Given the conditions, one possible rational function is \(\frac{1}{x}\). Here, \(P(x)=1\) is a constant polynomial (degree 0), and \(Q(x)=x\) is a polynomial of degree 1. The horizontal asymptote is the x-axis because the degree of the numerator is less than the degree of the denominator. The vertical asymptote is the y-axis because the denominator becomes zero when \(x=0\).

Key Concepts

asymptotes

When studying rational functions, understanding asymptotes is crucial. Asymptotes are lines that a graph approaches but never actually touches. They help us understand the behavior of a function as it moves towards infinity or a specific point.
There are two main types of asymptotes: horizontal and vertical.
Knowing their properties can help us correctly form rational functions matching given asymptotes.

horizontal asymptote

A horizontal asymptote helps define a function's end behavior.

When x becomes very large (positively or negatively), the graph of the function approaches a horizontal line. This line is the horizontal asymptote.

For a rational function given as \(\frac{P(x)}{Q(x)}\), where both \(P(x)\) and \(Q(x)\) are polynomials, the horizontal asymptote depends on their degrees:

• If the degree of \(P(x)\) is less than the degree of \(Q(x)\), the horizontal asymptote is \(y = 0\).
• If the degrees are equal, the asymptote is \(y = \frac{a}{b}\), where \(a\) and \(b\) are leading coefficients of \(P(x)\) and \(Q(x)\).
• If the degree of \(P(x)\) is greater than the degree of \(Q(x)\), there is no horizontal asymptote (the asymptote may be an oblique line).

In our exercise, the horizontal asymptote is the x-axis (\(y = 0\)). This tells us that the degree of the numerator must be less than the degree of the denominator.

vertical asymptote

A vertical asymptote is different from a horizontal one. It's a line the graph approaches but never crosses as the function's value becomes infinitely large or small. Vertical asymptotes typically arise at points where the denominator of a rational function equals zero.

For a rational function \(\frac{P(x)}{Q(x)}\), a vertical asymptote exists at x-values that make \(Q(x)\) equal to zero (provided these points don't cancel with zeros in the numerator).

• If \(Q(x)\) has a factor that becomes zero at, say, \(x = a\), then x = a is a vertical asymptote.

This was a key part of our example. Our rational function \(\frac{1}{x}\) has \(Q(x) = x\). Hence, it has a vertical asymptote at x = 0.

polynomials

Polynomials are fundamental expressions in algebra consisting of variables and coefficients. They are combined using addition, subtraction, multiplication, and non-negative integer exponents.
For rational functions, we express them as the ratio of two polynomials, \(\frac{P(x)}{Q(x)}\).

• \(P(x)\) is the numerator polynomial.
• \(Q(x)\) is the denominator polynomial.

The degrees of \(P(x)\) and \(Q(x)\) play a significant role in determining horizontal and vertical asymptotes.
For instance, in the function \(\frac{1}{x}\), both \(\frac{1}{x}\) and \(\frac{x^2}{x^3+1}\), the numerator polynomial has a lower degree than the denominator, leading to a horizontal asymptote at \(y=0\). The vertical asymptote occurs where the denominator is zero, highlighting the importance of understanding polynomials to form specific rational functions with desired asymptotic properties.