4.5: Rational Functions

Preview

Explain, in your own words, why we can’t divide by zero.

Discovering X-Intercepts, Domain, and Vertical Asymptotes of Rational Functions

Take a look at the two linear functions, f ( x )∧g ( x ) , graphed below. Graph the quotient f ( x )g ( x )

on the same set of

axes by filling out the corresponding chart.

x f ( x ) g ( x ) f ( x )g ( x )

-8-6-4-20246-4.5-3.5

Q: What are the equations of f ( x ) , g ( x ) ,∧f ( x )g ( x )

?

A: f ( x )=¿¿, g ( x )=¿¿, f ( x )g ( x )

=¿¿.

Graph f ( x )g ( x )

on your calculator and see if it matches the graph you drew.

Q: Take a look at our new function, f ( x )g ( x )

. What is the domain of this function (what values of x are allowed)?

What does this correspond to in terms of f and/or g? Why?A:

Example 1Find the domain of:

[A ] f ( x )= x2−25x−5

[B ] f (x )= x

x2−25[C ] f ( x )= x+5

x2+25

Q: Going back again to our f ( x )g ( x )

, what exactly happened at x=−4?!

A:

Let’s learn some vocab…

Definition 1. A rational function is a function that is a ____________________________ of two ____________________________.

That is, a rational function is h ( x )= f ( x )g (x )

,where f and g are polynomials and g(x )≠0.

Definition 2. The line x=a is a vertical asymptote of the graph of f if f(x) increases or decreases without bound as x approaches a.

The word asymptote comes from the Greek word asymptotes, meaning “not touching.”

Q: So, what was the vertical asymptote for f ( x )g ( x )

from the beginning of the lesson?

A:

How to find a vertical asymptote:

If h ( x )= f ( x )g (x )

is a rational function in which f(x) and g(x) have no common factors and a is a zero (or x-intercept)

of the denominator g(x), then x=a is a vertical asymptote of the graph of h.

Example 2Find the vertical asymptotes, if any, of:

[A ] h (x )= x

x2−1[B ] j ( x )= x−1

x2−1[C ]k ( x )= x+1

x2+1

Q: Can the denominator of a rational function have a zero at a but not have a vertical asymptote at x=a?A:

Hence, make sure you factor completely and divide any common factors before determining vertical asymptotes.

Q: List the x-intercept(s) of our favorite f ( x )g ( x )

. What does this correspond to in terms of f and/or g? Why?

A:

Example 3Find the x-intercepts of the following functions. Always make sure they’re in your domain.

[A ] f ( x )= x−2x+3 [B ] f (x )= x2

x−7 [C ] f ( x )= x

2+3x−4x2−5

Horizontal Asymptotes

If we have vertical asymptotes, we gotta have horizontal ones, too…

Definition 3. The line y=b is a horizontal asymptote of the graph of f if f(x) approaches b as x increases or decreases without bound.

While a function can have several vertical asymptotes it can have only one horizontal asymptote.Although a function can never intersect a vertical asymptote it may cross its horizontal asymptote.

Example 4Can you figure out which term(s) would have the most influence as x approaches −∞∨∞? (We write this as x→−∞ ,∨x→∞.) What does this tell us about our horizontal asymptotes?

[A ] f ( x )= 2 x3+4 x2−54 x3−8 x2+7 x

[B ] f (x )= 5x−8x2+3 x

[C ] f ( x )= x3+1x2−2 x+2

How to find horizontal asymptotes: (Compare the degrees of top [numerator] and bottom [denominator])

(1) Biggest on bottom, then y=____________ is the horizontal asymptote (like Example 4 ______).

(2) Biggest on top, then we have ______ horizontal asymptote (like Example 4 _____).

(3) Exponents are the same, then y=______________ is the horizontal asymptote, where a is the leading coefficient of the numerator; b is the leading coefficient of the denominator (like Example 4 _____).

Example 5Find the horizontal asymptote, if any, of:

[A ] f ( x )= 9 x2

3x2+1 [B ] g ( x )= 9x

3x2+1[C ]h ( x )= 9 x3

3x2+1

Oblique Asymptotes

Definition 4. Sometimes a line that is neither horizontal nor vertical is an asymptote. Such a line is called an oblique asymptote or a slant asymptote.

Q: What would be another way to write 2x2−3 x−1x−2

?

What happens to the remainder as x→−∞ or x→∞?A:

You can remember this

by BOBO BOTN

EATSDC

Example 6

State the oblique asymptote of f ( x )=2 x2−3 x−1x−2

.

There can be only __________ horizontal asymptote or __________ oblique asymptote and never both.Note: An asymptote is NOT part of the graph of the function.

Putting It All Together

Example 7State whether the following rational functions have a horizontal asymptote, oblique, or neither. If they have a horizontal or oblique asymptote, state what it is.

[A ] f ( x )=3x2+2 x4 x2

[B ] f (x )=2 x2+3x−5

[C ] f ( x )= x4+3 xx+1

Example 8

Find all asymptotes and intercepts of f ( x )= x+5x2−4 x−5

. State the domain. Graph the function.

You can remember this

by BOBO BOTN

EATSDC

Occurrence of Lines as Asymptotes of Rational FunctionsFor a rational function h ( x )=f (x) /g( x), where f and g have no common factors other than constants:

Vertical asymptotes occur at any x-values that make the denominator ________. The x-axis (or y=0) is the horizontal asymptote when the degree of the numerator is __________________

the degree of the denominator. A horizontal asymptote other than the x-axis occurs when the degree of the numerator is ____________

the degree of the denominator. These horizontal asymptotes take the form y=¿¿ where a and b are the leading coefficients of the numerator and denominator, respectively.

An oblique asymptote occurs when the degree of the numerator is _____ greater than the degree of the denominator.

Other Important Pieces

Domain restrictions are found by setting the _____________________ equal to ________. X-intercepts occur when the ________________________ is equal to __________. (Always check domain!) Y-intercepts can be found by computing ___________.

Example 9

Find all asymptotes and intercepts of f ( x )= x2−6 x+8x+2

. State the domain. Graph the function.

Example 10

The function N ( t )=0.8t+12004 t+7

,t ≥15, gives the body concentration N(t), in parts per million, of a certain dosage

of medicine after t hours.[A] What does N(t) approach as t goes to ∞ (would this be a horizontal or vertical asymptote)?

[B] Explain the meaning of part [A] in terms of the application. Does the concentration ever decrease to zero?

Q: What’s a rational function?A: