1. 3.6: Rational Functions and Their Graphs
Rational functions are quotients of polynomial functions. This means that rational functions can be expressed as
where p(x) and q(x) are polynomial functions and q(x)  0. The domain of a rational function is the set of all real numbers except the x-values that make the denominator zero. For example, the domain of the rational function
is the set of all real numbers except 0, 2, and -5.
This is p(x).
This is q(x).

2. EXAMPLE: Finding the Domain of a Rational Function
3.6: Rational Functions and Their Graphs
EXAMPLE: Finding the Domain of a Rational Function
Find the domain of each rational function. a.
Solution Rational functions contain division. Because division by 0 is undefined, we must exclude from the domain of each function values of x that cause the polynomial function in the denominator to be 0.
a. The denominator of is 0 if x = 3. Thus, x cannot equal 3. The domain of f consists of all real numbers except 3, written {x | x  3}.
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3. EXAMPLE: Finding the Domain of a Rational Function
3.6: Rational Functions and Their Graphs
EXAMPLE: Finding the Domain of a Rational Function
Find the domain of each rational function. a.
Solution Rational functions contain division. Because division by 0 is undefined, we must exclude from the domain of each function values of x that cause the polynomial function in the denominator to be 0.
b. The denominator of is 0 if x = -3 or x = 3. Thus, the domain of g consists of all real numbers except -3 and 3, written {x | x  - {x | x  -3, x  3}.
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4. EXAMPLE: Finding the Domain of a Rational Function
3.6: Rational Functions and Their Graphs
EXAMPLE: Finding the Domain of a Rational Function
Find the domain of each rational function. a.
Solution Rational functions contain division. Because division by 0 is undefined, we must exclude from the domain of each function values of x that cause the polynomial function in the denominator to be 0.
c. No real numbers cause the denominator of to equal zero. The domain of h consists of all real numbers.

5. Rational Functions Arrow Notation
3.6: Rational Functions and Their Graphs
Rational Functions
Unlike the graph of a polynomial function, the graph of the reciprocal function has a break in it and is composed of two distinct branches. We use a special arrow notation to describe this situation symbolically:
Arrow Notation
Symbol Meaning
x  a + x approaches a from the right.
x  a - x approaches a from the left.
x   x approaches infinity; that is, x increases without bound.
x  -  x approaches negative infinity; that is, x decreases without bound.

6. Vertical Asymptotes of Rational Functions
3.6: Rational Functions and Their Graphs
Vertical Asymptotes of Rational Functions
Definition of a Vertical Asymptote
The line x = a is a vertical asymptote of the graph of a function f if f (x) increases or decreases without bound as x approaches a.
f (x)   as x  a f (x)   as x  a - f a y x
x = a f a y x
x = a
Thus, f (x)   or f(x)  -  as x approaches a from either the left or the right.
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7. Vertical Asymptotes of Rational Functions
3.6: Rational Functions and Their Graphs
Vertical Asymptotes of Rational Functions
Definition of a Vertical Asymptote
The line x = a is a vertical asymptote of the graph of a function f if f (x) increases or decreases without bound as x approaches a.
f (x)  -  as x  a + f (x)  -  as x  a - f a y x
x = a
x = a f a y x
Thus, f (x)   or f(x)  -  as x approaches a from either the left or the right.

8. Vertical Asymptotes of Rational Functions
3.6: Rational Functions and Their Graphs
Vertical Asymptotes of Rational Functions
If the graph of a rational function has vertical asymptotes, they can be located by using the following theorem.
Locating Vertical Asymptotes
If is a rational function in which p(x) and q(x) have no common factors and a is a zero of q(x), the denominator, then x = a is a vertical asymptote of the graph of f.

9. Horizontal Asymptotes of Rational Functions
3.6: Rational Functions and Their Graphs
Horizontal Asymptotes of Rational Functions
A rational function may have several vertical asymptotes, but it can have at most one horizontal asymptote.
Definition of a Horizontal Asymptote
The line y = b is a horizontal asymptote of the graph of a function f if f (x) approaches b as x increases or decreases without bound. f y x
y = b x y f
y = b f y x
y = b
f (x)  b as x   f (x)  b as x   f (x)  b as x  

10. Horizontal Asymptotes of Rational Functions
3.6: Rational Functions and Their Graphs
Horizontal Asymptotes of Rational Functions
If the graph of a rational function has a horizontal asymptote, it can be located by using the following theorem.
Locating Horizontal Asymptotes
Let f be the rational function given by
The degree of the numerator is n. The degree of the denominator is m.
If n < m, the x-axis is the horizontal asymptote of the graph of f.
If n = m, the line y = is the horizontal asymptote of the graph of f.
If n > m, the graph of f has no horizontal asymptote.

11. f (-x) = f (x): y-axis symmetry f (-x) = -f (x): origin symmetry
3.6: Rational Functions and Their Graphs
Strategy for Graphing a Rational Function
Suppose that
where p(x) and q(x) are polynomial functions with no common factors.
1. Determine whether the graph of f has symmetry.
f (-x) = f (x): y-axis symmetry f (-x) = -f (x): origin symmetry
2. Find the y-intercept (if there is one) by evaluating f (0). 3. Find the x-intercepts (if there are any) by solving the equation p(x) = Find any vertical asymptote(s) by solving the equation q (x) = Find the horizontal asymptote (if there is one) using the rule for determining the horizontal asymptote of a rational function. 6. Plot at least one point between and beyond each x-intercept and vertical asymptote. 7. Use the information obtained previously to graph the function between and beyond the vertical asymptotes.

12. EXAMPLE: Graphing a Rational Function
3.6: Rational Functions and Their Graphs
EXAMPLE: Graphing a Rational Function
Solution
Step 1 Determine symmetry: f (-x) = = = f (x):
Symmetric with respect to the y-axis.
Step 2 Find the y-intercept: f (0) = = 0: y-intercept is 0.
Step 3 Find the x-intercept: 3x2 = 0, so x = 0: x-intercept is 0.
Step 4 Find the vertical asymptotes: Set q(x) = 0.
x2 - 4 = Set the denominator equal to zero.
x2 = 4
x = 2
Vertical asymptotes: x = -2 and x = 2.
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13. EXAMPLE: Graphing a Rational Function
3.6: Rational Functions and Their Graphs
EXAMPLE: Graphing a Rational Function
Solution
Step 5 Find the horizontal asymptote: y = 3/1 = 3.
Step 6 Plot points between and beyond the x-intercept and the vertical asymptotes. With an x-intercept at 0 and vertical asymptotes at x = 2 and x = -2, we evaluate the function at -3, -1, 1, 3, and 4. -5 -4 -3 -2 -1 1 2 3 4 5 7 6
Vertical asymptote: x = 2
Vertical asymptote: x = -2
Horizontal asymptote: y = 3
x-intercept and y-intercept x -3 -1 1 3 4
f(x) =
The figure shows these points, the y-intercept, the x-intercept, and the asymptotes.
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14. EXAMPLE: Graphing a Rational Function
3.6: Rational Functions and Their Graphs
EXAMPLE: Graphing a Rational Function
Solution
Step 7 Graph the function. The graph of f (x) = 2x is shown in the figure. The y-axis symmetry is now obvious. -5 -4 -3 -2 -1 1 2 3 4 5 7 6
Vertical asymptote: x = 2
Vertical asymptote: x = -2
Horizontal asymptote: y = 3
x-intercept and y-intercept -5 -4 -3 -2 -1 1 2 3 4 5 7 6
x = -2
y = 3
x = 2

15. EXAMPLE: Finding the Slant Asymptote of a Rational Function
3.6: Rational Functions and Their Graphs
EXAMPLE: Finding the Slant Asymptote of a Rational Function
Find the slant asymptotes of f (x) =
Solution Because the degree of the numerator, 2, is exactly one more than the degree of the denominator, 1, the graph of f has a slant asymptote. To find the equation of the slant asymptote, divide x - 3 into x2 - 4x - 5: 3
Remainder
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16. EXAMPLE: Finding the Slant Asymptote of a Rational Function
3.6: Rational Functions and Their Graphs
EXAMPLE: Finding the Slant Asymptote of a Rational Function
Find the slant asymptotes of f (x) =
Solution The equation of the slant asymptote is y = x - 1. Using our strategy for graphing rational functions, the graph of f (x) = is shown. -2 -1 4 5 6 7 8 3 2 1 -3
Vertical asymptote:
x = 3
Slant asymptote:
y = x - 1