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Key equations

Rational Function f ( x ) = P ( x ) Q ( x ) = a p x p + a p 1 x p 1 + ... + a 1 x + a 0 b q x q + b q 1 x q 1 + ... + b 1 x + b 0 ,   Q ( x ) 0

Key concepts

  • We can use arrow notation to describe local behavior and end behavior of the toolkit functions f ( x ) = 1 x and f ( x ) = 1 x 2 . See [link] .
  • A function that levels off at a horizontal value has a horizontal asymptote. A function can have more than one vertical asymptote. See [link] .
  • Application problems involving rates and concentrations often involve rational functions. See [link] .
  • The domain of a rational function includes all real numbers except those that cause the denominator to equal zero. See [link] .
  • The vertical asymptotes of a rational function will occur where the denominator of the function is equal to zero and the numerator is not zero. See [link] .
  • A removable discontinuity might occur in the graph of a rational function if an input causes both numerator and denominator to be zero. See [link] .
  • A rational function’s end behavior will mirror that of the ratio of the leading terms of the numerator and denominator functions. See [link] , [link] , [link] , and [link] .
  • Graph rational functions by finding the intercepts, behavior at the intercepts and asymptotes, and end behavior. See [link] .
  • If a rational function has x -intercepts at x = x 1 , x 2 , , x n , vertical asymptotes at x = v 1 , v 2 , , v m , and no x i = any  v j , then the function can be written in the form
    f ( x ) = a ( x x 1 ) p 1 ( x x 2 ) p 2 ( x x n ) p n ( x v 1 ) q 1 ( x v 2 ) q 2 ( x v m ) q n

    See [link] .

Section exercises

Verbal

What is the fundamental difference in the algebraic representation of a polynomial function and a rational function?

The rational function will be represented by a quotient of polynomial functions.

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What is the fundamental difference in the graphs of polynomial functions and rational functions?

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If the graph of a rational function has a removable discontinuity, what must be true of the functional rule?

The numerator and denominator must have a common factor.

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Can a graph of a rational function have no vertical asymptote? If so, how?

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Can a graph of a rational function have no x -intercepts? If so, how?

Yes. The numerator of the formula of the functions would have only complex roots and/or factors common to both the numerator and denominator.

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Algebraic

For the following exercises, find the domain of the rational functions.

f ( x ) = x + 1 x 2 1

All reals  x 1 ,   1

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f ( x ) = x 2 + 4 x 2 2 x 8

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f ( x ) = x 2 + 4 x 3 x 4 5 x 2 + 4

All reals  x 1 ,   2 ,   1 ,   2

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For the following exercises, find the domain, vertical asymptotes, and horizontal asymptotes of the functions.

f ( x ) = 2 5 x + 2

V.A. at x = 2 5 ; H.A. at y = 0 ; Domain is all reals x 2 5

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f ( x ) = x x 2 + 5 x 36

V.A. at x = 4 ,   9 ; H.A. at y = 0 ; Domain is all reals x 4 ,   9

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f ( x ) = 3 + x x 3 27

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f ( x ) = 3 x 4 x 3 16 x

V.A. at x = 0 ,   4 ,   4 ; H.A. at y = 0 ; Domain is all reals x 0 , 4 ,   4

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f ( x ) = x 2 1 x 3 + 9 x 2 + 14 x

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f ( x ) = x + 5 x 2 25

V.A. at x = 5 ; H.A. at y = 0 ; Domain is all reals x 5 , 5

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f ( x ) = x 4 x 6

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f ( x ) = 4 2 x 3 x 1

V.A. at x = 1 3 ; H.A. at y = 2 3 ; Domain is all reals x 1 3 .

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For the following exercises, find the x - and y -intercepts for the functions.

f ( x ) = x x 2 x

none

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f ( x ) = x 2 + 8 x + 7 x 2 + 11 x + 30

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f ( x ) = x 2 + x + 6 x 2 10 x + 24

x -intercepts none,  y -intercept  ( 0 , 1 4 )

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f ( x ) = 94 2 x 2 3 x 2 12

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For the following exercises, describe the local and end behavior of the functions.

Questions & Answers

For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
Shakeena Reply
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
Rhudy Reply
what is a complex number used for?
Drew Reply
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
Is there any rule we can use to get the nth term ?
Anwar Reply
how do you get the (1.4427)^t in the carp problem?
Gabrielle Reply
A hedge is contrusted to be in the shape of hyperbola near a fountain at the center of yard.the hedge will follow the asymptotes y=x and y=-x and closest distance near the distance to the centre fountain at 5 yards find the eqution of the hyperbola
ayesha Reply
A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour. To the nearest hour, what is the half-life of the drug?
Sandra Reply
Find the domain of the function in interval or inequality notation f(x)=4-9x+3x^2
prince Reply
hello
Jessica Reply
Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the high temperature of ?105°F??105°F? occurs at 5PM and the average temperature for the day is ?85°F.??85°F.? Find the temperature, to the nearest degree, at 9AM.
Karlee Reply
if you have the amplitude and the period and the phase shift ho would you know where to start and where to end?
Jean Reply
rotation by 80 of (x^2/9)-(y^2/16)=1
Garrett Reply
thanks the domain is good but a i would like to get some other examples of how to find the range of a function
bashiir Reply
what is the standard form if the focus is at (0,2) ?
Lorejean Reply
a²=4
Roy Reply
Practice Key Terms 5

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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