2.6 Introduction to Rational Functions

A rational function is a function of the form f(x) = N(x)/D(x), where N andD are both polynomials and D(x) 0 . The domain of f is all real x’s except x values that give 0 in the denominator. N(x) and D(x) should have no common factors.

values.- xexcludedany near f ofbehavior thediscuss and x

1f(x) ofdomain theFind

Example 1

x -1 -1/2 -1/10 -1/100 -1/1000 0

f(x) -1 -2 -10 -100 -1000

Domain is all reals except x=0

x 0 1/1000 1/100 1/10 1/2 1

f(x) 1000 100 10 2 1

Plug in x value to find y value.

This table shows x approaching 0(the excluded x value) from the left.

This table shows x approaching 0 from the right.

values.- xexcludedany near f ofbehavior thediscuss and x

1f(x) ofdomain theFind

x -1 -1/2 -1/10 -1/100 -1/1000 0

f(x) -1 -2 -10 -100 -1000

x 0 1/1000 1/100 1/10 1/2 1

f(x) 1000 100 10 2 1

Plot the sets of ordered pairs and this is the graph that you get.

Note that as x approaches 0 from the left, f(x) decreases without bound. In contrast, as x approaches 0 from the right, f(x) increases without bound.

What do you notice about the line x=0 and the graph of f ???

x

f(x)

xf(x)

Remember that the domain is all reals except x=0.

The graph never touches the line. This line is a vertical asymptote.

Vertical and Horizontal Asymptotes• 1. The line x = a is a vertical asymptote of the

graph of f if f(x) as x a, either from the right or left.

• 2. The line y = b is a horizontal asymptote of the graph of f if f(x) b as x

Asymptotes of Rational Functions Rules:

D(x)

N(x)f(x)function rational a be fLet

reelowerxb

reelowerxam

m

nn

deg

deg

where N(x) and D(x) have no common factors.

1. The graph of f has vertical asymptotes at the zeros of D(x).

2. The graph of f has at most one horizontal asymptote determined by comparing the degrees of N(x) and D(x).

a. If n<m, the line y=0 ( the x-axis) is a horizontal asymptote.

b. If n=m, the line y=an/bm is a horizontal asymptote.

c. If n>m , the graph of f has no horizontal asymptote.

• Slant Asymptotes– Only occur when the degree of the top is 1

more than the degree of the bottom. – The S.A. is derived by dividing the top by the

bottom (long division or synthetic) and ignoring the remainder.

Asymptotes of Rational Functions Rules:

a. If n<m, the line y=0 ( the x-axis) is a horizontal asymptote.b. If n=m, the line y=an/bm is a horizontal asymptote.c. If n>m , the graph of f has no horizontal asymptote.

Ex. 2 Find the Horizontal Asymptotes for each of the following functions.

13

2)(

2 x

xxf n<m therefore the horizontal asymptote is

y=0.

a. If n<m, the line y=0 ( the x-axis) is a horizontal asymptote.b. If n=m, the line y=an/bm is a horizontal asymptote.c. If n>m , the graph of f has no horizontal asymptote.

Ex. 2 Find the Horizontal Asymptotes for each of the

following functions.

13

2)(

2

2

x

xxg n=m therefore, y=2/3 is the horizontal

asymptote.

a. If n<m, the line y=0 ( the x-axis) is a horizontal asymptote.b. If n=m, the line y=an/bm is a horizontal asymptote.c. If n>m , the graph of f has no horizontal asymptote.

Ex. 2 Find the Horizontal Asymptotes for each of the following functions.

13

2)(

2

3

x

xxh n>m, therefore there is no horizontal

asymptote.

Although this graph does not have a horizontal asymptote it does have a slant or oblique asymptote – the line y=2/3 x.

Ex. 3 For the function f, find a) the domain of f, b) the vertical asymptotes of f, and c) the horizontal asymptote of f.

54

273)(

3

23

x

xxxf

08.14

5

4

5

54

054x-

3

3

3

3

x

x

x

08.14

5except x

reals all isdomain The

3

a) Set denominator =0 and solve.

b) The graph of f has vertical asymptotes at the zeros of D(x). Therefore the vertical asymptote of f is

08.14

53 x

c) If n=m, the line y=an/bm is a horizontal asymptote. Therefore, the horizontal asymptote is

4

3y

54

273)(

3

23

x

xxxf

08.14

5except x

reals all isdomain The

3

08.14

5

is asymptote verticalThe

3 x

4

3

is asymptote horizontal The

y

Ex. 4 A Graph with Two Horizontal AsymptotesA function that is not rational can have two horizontal asymptotes-one

to the left and one to the right. For instance, the graph of

2

10)(

x

xxf

0 x,2x

10x

0 x,2

10)(

x

xxf HA y = -1

HA y = 1

Find the following if possible: domain, vertical asymptote, horizontal asymptote, slant asymptote.

Find the following if possible: domain, vertical asymptote, horizontal asymptote, slant asymptote.

1

2)(

1

2)(

2

2

x

xxxf

xxf

Ultraviolent Radiation

• For a person with sensitive skin, the amount of time T (in hours) the person can be exposed to the sun with mininal burning can be modeled by

where s is the Sunsor Scale Reading. The Sunsor Scale is based on the level of intensity of UVB rays.

0.37 23.8,0 120

sT s

s

Homework

• Page 152-155

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