SECTION 3.2SECTION 3.2

RATIONAL FUNCTIONSRATIONAL FUNCTIONS

RATIONAL FUNCTIONSRATIONAL

FUNCTIONS

Rational functions take on Rational functions take on the form:the form:

f(x) = p(x)

q(x)

Where p(x) and q(x) are Where p(x) and q(x) are polynomials and q(x) polynomials and q(x) 0 0

EXAMPLES:EXAMPLES:

f(x) = 2x - 3

x + 1 g(x) =

x + 3x - 1

x + 1

3

2

h(x) = x - x - 6

x - 4

2

2

DOMAIN OF RATIONAL

FUNCTIONS

DOMAIN OF RATIONAL

FUNCTIONS ANY VALUE WHICH ZEROS ANY VALUE WHICH ZEROS

OUT THE DENOMINATOR OUT THE DENOMINATOR MUST BE EXCLUDED FROM MUST BE EXCLUDED FROM THE DOMAIN.THE DOMAIN.

Do Example 1Do Example 1

GRAPHS OF RATIONAL

FUNCTIONS

GRAPHS OF RATIONAL

FUNCTIONS ASYMPTOTEASYMPTOTE - a line which - a line which

the graph will approach but the graph will approach but will never reach.will never reach.

HORIZONTAL ASYMPTOTESHORIZONTAL ASYMPTOTES

When the degree in the When the degree in the numerator is numerator is equalequal to the to the degree in the denominator.degree in the denominator.

Example: GraphExample: Graph

f(x) = 3x - 4

x + 1

2

2

By studying the graph, we can By studying the graph, we can describe what is happening using describe what is happening using the following symbols:the following symbols:

As x As x , f(x) , f(x) 3 3 As x As x - - , f(x) , f(x) 3 3 We say f(x) has a horizontal We say f(x) has a horizontal

asymptote of y = 3.asymptote of y = 3.

To see the algebraic reasoning To see the algebraic reasoning behind this, divide the rational behind this, divide the rational expression through by xexpression through by x22 and and examine what happens as x gets examine what happens as x gets huge.huge.

3xx

- 4x

xx

+ 1x

2

2 2

2

2 2

The numerator goes to 3 and the The numerator goes to 3 and the denominator goes to 1.denominator goes to 1.

EXAMPLE:EXAMPLE:

Graph: f(x) = 4x - 8x

x + 1 on -10 x 10

2

2

4xx

- 8xx

xx

+ 1x

2

2 2

2

2 2

Again, algebraically, we can see that Again, algebraically, we can see that the numerator goes to 4 and the the numerator goes to 4 and the denominator goes to 1 as x gets huge. denominator goes to 1 as x gets huge. Thus, this function has a horizontal Thus, this function has a horizontal asymptote y = 4.asymptote y = 4.

We can get even more specific with We can get even more specific with our symbols when describing the our symbols when describing the graph of the function and say the graph of the function and say the following:following:

As x As x , f(x) , f(x) 4 4 - -

As x As x - - , f(x) , f(x) 4 4++

QUESTION:QUESTION:

Can you find an easy way of Can you find an easy way of looking at the symbolic form looking at the symbolic form of a rational function in which of a rational function in which the degree in the numerator is the degree in the numerator is equalequal to the degree in the to the degree in the denominator to find the denominator to find the horizontal asymptote?horizontal asymptote?

f(x) = 3x - 4

x + 1

2

2

f(x) = 4x - 8x

x + 1

2

2

Answer:Answer:Just divide the leading Just divide the leading coefficient in the numerator by coefficient in the numerator by the leading coefficient in the the leading coefficient in the denominator.denominator.

EXAMPLE:EXAMPLE:

Determine the equation of the Determine the equation of the horizontal asymptote for the horizontal asymptote for the graph of :graph of :

f(x) = 3x + 2x + 5

2x + x + 9

6 5

6 4

Horizontal Asymptote: y = 3/2Horizontal Asymptote: y = 3/2

VERTICAL ASYMPTOTES:

VERTICAL ASYMPTOTES:

Vertical lines are always given Vertical lines are always given by an equation in the form x = c. by an equation in the form x = c. Here, the graph of the rational Here, the graph of the rational function will approach a vertical function will approach a vertical line, yet never quite reach it. line, yet never quite reach it. This means that this is a value x This means that this is a value x will never equal.will never equal.

Vertical asymptotes are nothing Vertical asymptotes are nothing more than domain restrictions, or more than domain restrictions, or values for the variable that will values for the variable that will cause the denominator to equal 0.cause the denominator to equal 0.

EXAMPLE:EXAMPLE:

f(x) = 1

x - 2

x = 2 is not in the domain of f(x) x = 2 is not in the domain of f(x) because replacing x with 2 because replacing x with 2 would cause the denominator to would cause the denominator to equal 0. Graph f(x).equal 0. Graph f(x).

Here, we can describe what is Here, we can describe what is happening to the graph of the happening to the graph of the function by using the function by using the following language:following language:

As x As x 2 2 - - , f(x) , f(x) As x As x 22+ + ,, f(x) f(x)

Thus, the vertical asymptote for Thus, the vertical asymptote for this function is x = 2.this function is x = 2.

Is this the only asymptote for this Is this the only asymptote for this function?function?

No! This function also has a No! This function also has a horizontal asymptote given by the horizontal asymptote given by the equation y = 0equation y = 0

As x gets huge, the numerator As x gets huge, the numerator goes to zero and the denominator goes to zero and the denominator goes to 1.goes to 1.

1x

xx

- 2x

In fact, any rational function in which the degree in the In fact, any rational function in which the degree in the numerator is less than the degree in the denominator will have a numerator is less than the degree in the denominator will have a horizontal asymptote of y = 0 (or the x-axis).horizontal asymptote of y = 0 (or the x-axis).

EXAMPLE:EXAMPLE:

f(x) = 3x + 5

x - 4x + 4

2

2

We can either examine the graph to We can either examine the graph to determine the asymptotes, or we can determine the asymptotes, or we can study the symbolic form, using the study the symbolic form, using the tools we’ve learned.tools we’ve learned.

The vertical asymptotes will be the The vertical asymptotes will be the domain restrictions. What values will domain restrictions. What values will zero out the denominator?zero out the denominator?

f(x) = 3x + 5

(x - 2)

2

2

Thus, the only vertical asymptote is Thus, the only vertical asymptote is x = 2.x = 2.

f(x) = 3x + 5

x - 4x + 4

2

2

Now, for the horizontal Now, for the horizontal asymptotes.asymptotes.

The degree in the numerator is The degree in the numerator is equal to the degree in the equal to the degree in the denominator.denominator.

Thus, the horizontal asymptote is Thus, the horizontal asymptote is y = 3. Graph the function.y = 3. Graph the function.

EXAMPLE:EXAMPLE:

Determine all asymptotes:Determine all asymptotes:

f(x) = 4x

x + 2x - 8

2

2

f(x) = 4x

(x + 4)(x - 2)

2

H.A. y = 4H.A. y = 4 V.A.V.A. x = - 4x = - 4x = 2x = 2

SLANT ASYMPTOTESSLANT ASYMPTOTES

When the degree in the numerator When the degree in the numerator is exactly one more than the is exactly one more than the degree in the denominator.degree in the denominator.

f(x) = x - 3x + 6

x - 2

2

f(x) = x - 3x + 6

x - 2

2

To determine the slant To determine the slant asymptote, we simply do the asymptote, we simply do the division implied by the division implied by the fraction line. We can use fraction line. We can use synthetic division.synthetic division.

f(x) = x - 3x + 6

x - 2

2

22 11 - 3- 3 6 6

11

22

- 1- 1

- 2- 2

44

Thus, f(x) can be written as:Thus, f(x) can be written as:

f(x) = x - 3x + 6

x - 2 = x - 1 +

4

x - 2

2

f(x) = x - 3x + 6

x - 2 = x - 1 +

4

x - 2

2

As x gets huge, the As x gets huge, the fractional part tends toward fractional part tends toward 0 and the entire function 0 and the entire function tends toward the linear tends toward the linear function y = x - 1.function y = x - 1.

Graph the function.Graph the function.

EXAMPLE:EXAMPLE:

Determine all asymptotes Determine all asymptotes of the function below:of the function below:

f(x) = 2x + 3x

x + x - 6

3 2

2

Vertical Asymptotes:Vertical Asymptotes:

x = - 3x = - 3 x = 2x = 2

Slant Asymptote:Slant Asymptote:

x + x - 6 2x + 3x + 0x + 02 3 2

2x2x

2x2x33 + 2x + 2x22 - 12x - 12x

xx22 + 12x + 12x

+ 1+ 1

xx22 + x - 6 + x - 6 11x + 611x + 6

f(x) = 2x + 3x

x + x - 6 = 2x + 1 +

11x + 6

x + x - 6

3 2

2 2

Slant Asymptote:Slant Asymptote:

y = 2x + 1y = 2x + 1

Graph the functionGraph the function

CONCLUSION OF SECTION 3.2CONCLUSION OF SECTION 3.2