Chapter 7

Polynomial and Rational Functions with Applications

Section 7.2

Section 7.2Rational Functions and Their Graphs

• Definition of a Rational Function

• Identifying Graphs of Rational Functions

• Asymptotes and Holes of Rational Functions

Definition of a Rational Function

A rational function f is a quotient of polynomial functions. That is,

where P(x) and Q(x) are polynomials and Q(x) 0.

)x(Q)x(P

)x(f

Determine the domain of each rational function.

a.

Since division by zero is undefined, the domain of a rational function is the set of all real numbers, except those values of x that would yield a zero denominator. We solve x2 – 9 = 0.

(x + 3)(x – 3) = 0

x = –3 or x = 3

Hence, for the given function, x –3, 3

b.

92x

x7)x(f

Since x2 0 for all x-values, there is no input that would make x2 + 49 equal to zero. There are no restrictions for this rational function, and the domain is the set of all real numbers.

492x

52x)x(f

Graphs of Rational Functions

Recall the basic function

Domain: (–∞, 0) U (0, ∞)

Range: (–∞, 0) U (0, ∞)

Since x ≠ 0, the graph of this function will approach, but not intersect the y-axis, Therefore, the y-axis (or the line x = 0) is a vertical asymptote of this rational function, as displayed on the graph.

As x 0– (x approaches zero from the left), y –∞.As x 0+ (x approaches zero from the right), y ∞.

.x1

)x(f

Vertical Asymptote of a Rational Function

Let be a rational function such that P(x) and

Q(x) have no common factors. The line x = k is a vertical asymptote of the graph of the function if Q(x) = 0.

To find the vertical asymptote of the graph of a given rational function f(x), set the denominator equal to 0 and solve for x. The resulting x-value(s) will not be part of the function's domain, thus a vertical asymptote will occur at such x-value(s).

Notes:A rational function may have many vertical asymptotes, or none at all.

The graph of a rational function will never cross a vertical asymptote.

 

)x(Q)x(P

)x(f

Horizontal Asymptote of a Rational Function

Let be a rational function. The line y = k is a

horizontal asymptote of the graph of the function if f(x) approaches k as x –∞, or as x ∞.

Notes:A rational function may have at most one horizontal asymptote.

The graph of a rational function may cross a horizontal asymptote.

 

)x(Q)x(P

)x(f

Finding a Horizontal Asymptote of a Rational Function

Let m represent the degree of the numerator. Let n represent the degree of the denominator.

1. If m < n, the line y = 0 (that is, the x-axis) is the horizontal asymptote of f.

2. If m = n, the line y = k is the horizontal asymptote of f, where k is the ratio of the leading coefficients. That is, k is found by dividing the leading coefficient of the numerator bythe leading coefficient of the denominator.

3. If m > n, the function does not have a horizontal asymptote.

 

For each of the following rational functions, (i) state the domain of the function, (ii) find the vertical asymptote(s), if any, and (iii) find the horizontal asymptote, if it exists.

a. b. c. 12x

2x3)x(f

a. Domain: (–∞, ∞) Vertical asymptote: None (there are no restrictions for x) Horizontal asymptote: y = 3/1 (ratio of the leading coefficients) b. Domain: (–∞, 0) U (0, ∞) Vertical asymptote: x = 0 (recall division by zero is undefined) Horizontal asymptote: None (degree of the numerator is larger than the degree of the denominator)

c. Domain: (–∞, –1) U (–1, 1) U (1, ∞) Vertical asymptotes: x = –1 and x = 1 Horizontal asymptote: y = 0 (degree of the numerator is smaller than the degree of the denominator)

x23x

)x(f

12x

2x)x(f

Horizontal Asymptote: Will the Graph Cross it?

Suppose a rational function f(x) has a horizontal asymptote at y = k. To determine algebraically if the graph will cross the asymptote, we can set the function equal to k and solve for x.

Example:

The graph of has a horizontal asymptote at y = 0.

Setting f(x) = 0, and solving for x we have

The graph will cross its horizontal asymptote at (–2, 0).

Note: If a rational function has a horizontal asymptote at y = k, and f(x) = k has no solution, then the graph of the rational function will not cross its asymptote.

12x2

6x3)x(f

)12x2(0)12x2(12x2

6x3

2x

06x3

Using your textbook, practice the problems assigned by your instructor to review the concepts from Section 7.2.