f x qx() - PVAMU Homex x = + f. The expression can be written as indicating that the graph may be...

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Rational function

A function ƒ of the form

( )( ) ,( )

p xxq x

=f

where p(x) and q(x) are polynomials, with q(x) ≠ 0, is called a rational function.

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Rational Function

Some examples of rational functions are 2

2 21 1 3 3 6( ) , ( ) , and ( )

2 5 3 8 16x x xx x x

x x x x x+ − −= = =

+ − + +f f f

Since any values of x such that q(x) = 0 are excluded from the domain of a rational function, this type of function often has a discontinuous graph, that is, a graph that has one or more breaks in it.

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The Reciprocal Function

The simplest rational function with a variable denominator is the reciprocal function, defined by

1( ) .xx

=f

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Domain: (–∞, 0) ∪ (0, ∞) Range: (–∞, 0) ∪ (0, ∞)

RECIPROCAL FUNCTION 1( )xx

=f

x y–2 –½

–1 –1

–½ –20 undefined

½ 21 12 ½

It is an odd function, and its graph is symmetric with respect to the origin.

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Example 2 GRAPHING A RATIONAL FUNCTION

Solution

Graph Give the domain and range.

2( ) .1

xx

=+

f

The expression can be written as indicating that the graph may be obtained by shifting the graph ofto the left 1 unit and stretchingit vertically by a factor of 2.

21x +12

1x +

1yx

=

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Example 2 GRAPHING A RATIONAL FUNCTION

Solution

Graph Give the domain and range.

2( ) .1

xx

=+

f

The horizontal shift affects the domain, which is now(–∞, –1) ∪ (–1, ∞) .The line x = –1 is the vertical asymptote, and the line y = 0(the x-axis) remains the horizontal asymptote. The range is still (–∞, 0) ∪ (0, ∞).

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Asymptotes

Let p(x) and q(x) define polynomials. For the rational function defined by written in lowest terms, and for real numbers a and b:

1. If |ƒ(x)| → ∞ as x → a, then the line is a vertical asymptote.

2. If ƒ(x) → b as |x| → ∞, then the line y = b is a horizontal asymptote.

( )( ) ,( )

p xxq x

=f

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Determining Asymptotes

To find the asymptotes of a rational function defined by a rational expression in lowest terms, use the following procedures.1. Vertical Asymptotes

Find any vertical asymptotes by setting the denominator equal to 0 and solving for x. If a is a zero of the denominator, then the line x = a is a vertical asymptote.

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2. Other AsymptotesDetermine any other asymptotes. Consider three possibilities:(a) If the numerator has lower degree

than the denominator, then there is a horizontal asymptote y = 0 (the x-axis).

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2. Other AsymptotesDetermine any other asymptotes. Consider three possibilities:(b) If the numerator and denominator have the

same degree, and the function is of the form

where an, bn ≠ 0,

then the horizontal asymptote has equation .n

n

ayb

=

0

0

( ) ,n

nn

n

a x axb x b

+ +=

+ +

f

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2. Other AsymptotesDetermine any other asymptotes. Consider three possibilities:(c) If the numerator is of degree exactly one

more than the denominator, then there will be an oblique (slanted) asymptote. To find it, divide the numerator by the denominator and disregard the remainder. Set the rest of the quotient equal to y to obtain the equation of the asymptote.

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Example 4 FINDING ASYMPTOTES OF GRAPHS OF RATIONAL FUNCTIONS

Solution To find the vertical asymptotes, set the denominator equal to 0 and solve.

a.

For each rational function ƒ, find all asymptotes.

1( )(2 1)( 3)

xxx x

+=− +

f

(2 1)( 3) 0x x− + =

2 1 0 or 3 0x x− = + = Zero-property

1 or 32

x x= = − Solve each equation.

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The equations of the vertical asymptotes are x = ½ and x = –3.To find the equation of the horizontal asymptote, divide each term by the greatest power of x in the expression. First, multiply the factors in the denominator.

21 1( )

(2 1)( 3) 2 5 3x xx

x x x x+ += =

− + + −f

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SolutionSet the denominator x – 3 = 0 equal to 0 to find that the vertical asymptote has equation x = 3. To find the horizontal asymptote, divide each term in the rational expression by x since the greatest power of xin the expression is 1.

b.


2 1( )3

xxx

+=−

f

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Solution Setting the denominator x – 2 equal to 0 shows that the vertical asymptote has equation x = 2. If we divide by the greatest power of x as before ( inthis case), we see that there is no horizontal asymptote because

c.


2 1( )2

xxx+=−

f

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We use synthetic division.

The result allows us to write the function as

2 1 0 12 4

1 2 5

5( ) 2 .2

x xx

= + +−

f

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For very large values of |x|, is close to 0, and the graph approaches the line y = x + 2. This line is an oblique asymptote (slanted, neither vertical nor horizontal) for the graph of the function.

52x −

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Steps for Graphing Functions

A comprehensive graph of a rational function exhibits these features:1. all x- and y-intercepts;2. all asymptotes: vertical, horizontal, and/or

oblique;3. the point at which the graph intersects its

nonvertical asymptote (if there is any such point);

4. the behavior of the function on each domain interval determined by the verticalasymptotes and x-intercepts.

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Example 5 GRAPHING A RATIONAL FUNCTION WITH THE x-AXIS AS HORIZONTAL ASYMPTOTE

Solution

Graph 21( ) .

2 5 3xx

x x+=

+ −f

Step 1 Since 2x2 + 5x – 3 = (2x – 1)(x + 3), from Example 4(a), the vertical asymptotes have equations x = ½ and x = –3.

Step 2 Again, as shown in Example 4(a), the horizontal asymptote is the x-axis.

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Solution

Graph 21( ) .

2 5 3xx

x x+=

+ −f

Step 3 The y-intercept is –⅓, since

2( ) .2( ) 5

1(

000 0

13 3)

+= =+

−−

f The y-intercept is the ratio of the constant terms.

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Solution

Graph 21( ) .

2 5 3xx

x x+=

+ −f

Step 4 The x-intercept is found by solving ƒ(x) = 0.

21

2 5 30x

x x+ =

+ −If a rational expression is equal to 0, then its numerator must equal 0.

1x = − The x-intercept is –1.

1 0x + =

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Solution

Graph 21( ) .

2 5 3xx

x x+=

+ −f

Step 5 To determine whether the graph intersects its horizontal asymptote, solve

(0) .0=f y-value of horizontal asymptote

Since the horizontal asymptote is the x-axis, the solution of this equation was found in Step 4. The graph intersects its horizontal asymptote at (–1, 0).

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Solution

Graph 21( ) .

2 5 3xx

x x+=

+ −f

Step 6 Plot a point in each of the intervals determined by the x-intercepts and vertical asymptotes, to get an idea of how the graph behaves in each interval.

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Solution

Graph 21( ) .

2 5 3xx

x x+=

+ −f

Step 7 Complete the sketch.

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Example 6 GRAPHING A RATIONAL FUNCTION THAT DOES NOT INTERSECT ITS HORIZONTAL ASYMPTOTE

Solution

Graph2 1( ) .

3xx

x+=−

f

Step 1 and 2 As determined in Example 4(b), the equation of the vertical asymptote is x = 3. The horizontal asymptote has equation y = 2.

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Solution

Graph2 1( ) .

3xx

x+=−

f

Step 3 ƒ(0) = –⅓, so the y-intercept is –⅓.

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Solution

Graph2 1( ) .

3xx

x+=−

f

Step 4 Solve ƒ(x) = 0 to find any x-intercepts.2

301x

x+ =−

If a rational expression is equal to 0, then its numerator must equal 0.2 1 0x + =

12

x = − x-intercept

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Solution

Graph2 1( ) .

3xx

x+=−

f

Step 5 The graph does not intersect its horizontal asymptote since ƒ(x) = 2 has no solution.

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Solution

Graph2 1( ) .

3xx

x+=−

f

Step 6 and 7 The points (–4, 1), (1, –3/2), and (6, 13/3) are on the graph and can be used to complete the sketch.

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Behavior of Graphs

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Example 8 GRAPHING A RATIONALFUNCTION WITH AN OBLIQUE ASYMPTOTE

Solution In Example 4, the vertical asymptote has equation x = 2, and the graph has an oblique asymptote with equation y = x + 2. Refer to the previous discussion to determine the behavior near the vertical asymptote x = 2.

Graph 2 1( ) .

2xxx+=−

f

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Solution The y-intercept is – ½ , and the graph has no x-intercepts since the numerator, x2 + 1, has no real zeros. The graph does not intersect its oblique asymptote because

Graph 2 1( ) .

2xxx+=−

f

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Solution

Graph 2 1( ) .

2xxx+=−

f

2 1 22

x xx+ = +−

has no solution. Using the y-intercept, asymptotes, the points andand the general behavior of the graph near its asymptotes leads to this graph.

174,2

21, ,3

− −

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Solution

Graph 2 1( ) .

2xxx+=−

f

f x qx() - PVAMU Homex x = + f. The expression can be written as indicating that the graph may be...

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