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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

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Practice Key Terms 5

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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