Rational Functions
Overview
A rational function is any function that can be written as the ratio of two polynomials. In other words, it is a function of the form:
\( f(x) = \frac{P(x)}{Q(x)} \)
where \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) \neq 0 \) because division by zero is undefined. Rational functions often involve simplifying the function, finding asymptotes, and analyzing its domain and range.
Key Features of Rational Functions
- Domain: The domain of a rational function consists of all real numbers except where the denominator is zero.
- Vertical Asymptotes: Vertical asymptotes occur where the denominator is zero and the numerator is nonzero.
- Horizontal Asymptotes: Horizontal asymptotes describe the behavior of the function as \( x \) approaches positive or negative infinity. They are determined by the degrees of the numerator and denominator polynomials.
- Intercepts: Rational functions can have both x- and y-intercepts, which can be found by setting the numerator equal to zero for x-intercepts and the denominator equal to zero for undefined values.
Example 1: Simplifying a Rational Function
Consider the following rational function:
\( f(x) = \frac{2x^2 + 4x}{4x^2 + 8x} \)
Step 1: Factor both the numerator and the denominator:
\( f(x) = \frac{2x(x + 2)}{4x(x + 2)} \)
Step 2: Cancel the common factor of \( (x + 2) \):
\( f(x) = \frac{2x}{4x} \)
Step 3: Simplify the expression:
\( f(x) = \frac{1}{2} \)
The simplified rational function is \( f(x) = \frac{1}{2} \).
Example 2: Analyzing Asymptotes
Consider the following rational function:
\( f(x) = \frac{x^2 - 4}{x^2 - 1} \)
Step 1: Factor both the numerator and the denominator:
\( f(x) = \frac{(x - 2)(x + 2)}{(x - 1)(x + 1)} \)
Step 2: Identify the vertical asymptotes:
Vertical asymptotes occur where the denominator is zero and the numerator is nonzero. In this case, the vertical asymptotes are at \( x = 1 \) and \( x = -1 \).
Step 3: Determine the horizontal asymptote:
Since the degrees of the numerator and denominator are both 2, the horizontal asymptote is found by dividing the leading coefficients. The leading coefficient of the numerator is 1, and the leading coefficient of the denominator is also 1. Therefore, the horizontal asymptote is:
\( y = 1 \)
Practice Questions
- Question 1: Simplify the following rational function:
\( f(x) = \frac{3x^2 - 6x}{6x^2 - 12x} \)
Solution
Step 1: Factor both the numerator and the denominator:
\( f(x) = \frac{3x(x - 2)}{6x(x - 2)} \)
Step 2: Cancel the common factor of \( (x - 2) \):
\( f(x) = \frac{3x}{6x} \)
Step 3: Simplify the expression:
\( f(x) = \frac{1}{2} \)
Therefore, the simplified rational function is \( f(x) = \frac{1}{2} \).
- Question 2: Find the vertical asymptotes of the following rational function:
\( f(x) = \frac{x^2 - 9}{x^2 - 4} \)
Solution
Step 1: Factor both the numerator and the denominator:
\( f(x) = \frac{(x - 3)(x + 3)}{(x - 2)(x + 2)} \)
Step 2: Identify the vertical asymptotes:
Vertical asymptotes occur where the denominator is zero and the numerator is nonzero. In this case, the vertical asymptotes are at \( x = 2 \) and \( x = -2 \).
- Question 3: Find the horizontal asymptote of the following rational function:
\( f(x) = \frac{3x^2 + 5x}{2x^2 - x + 1} \)
Solution
Step 1: Identify the degrees of the numerator and denominator. Both the numerator and denominator have degree 2.
Step 2: The horizontal asymptote is determined by dividing the leading coefficients. The leading coefficient of the numerator is 3, and the leading coefficient of the denominator is 2.
\( y = \frac{3}{2} \)
Therefore, the horizontal asymptote is \( y = \frac{3}{2} \).
- Question 4: Find the domain of the following rational function:
\( f(x) = \frac{x + 1}{x^2 - 4} \)
Solution
Step 1: Set the denominator equal to zero and solve for \( x \):
\( x^2 - 4 = 0 \)
\( x^2 = 4 \)
\( x = \pm 2 \)
Step 2: The function is undefined at \( x = 2 \) and \( x = -2 \). Therefore, the domain of the function is all real numbers except \( x = 2 \) and \( x = -2 \).
The domain is \( x \in \mathbb{R}, x \neq 2, x \neq -2 \).