<< Chapter < Page Chapter >> Page >

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.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

Algebraic

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

f ( x ) = x + 1 x 2 1

All reals  x 1 ,   1

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

All reals  x 1 ,   2 ,   1 ,   2

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

f ( x ) = 3 + x x 3 27

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

f ( x ) = x + 5 x 2 25

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

Got questions? Get instant answers now!

f ( x ) = x 4 x 6

Got questions? Get instant answers now!

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 .

Got questions? Get instant answers now!

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

f ( x ) = x x 2 x

none

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Questions & Answers

the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
Kc Reply
1+cos²A/cos²A=2cosec²A-1
Ramesh Reply
test for convergence the series 1+x/2+2!/9x3
success Reply
a man walks up 200 meters along a straight road whose inclination is 30 degree.How high above the starting level is he?
Lhorren Reply
100 meters
Kuldeep
Find that number sum and product of all the divisors of 360
jancy Reply
answer
Ajith
exponential series
Naveen
what is subgroup
Purshotam Reply
Prove that: (2cos&+1)(2cos&-1)(2cos2&-1)=2cos4&+1
Macmillan Reply
e power cos hyperbolic (x+iy)
Vinay Reply
10y
Michael
tan hyperbolic inverse (x+iy)=alpha +i bita
Payal Reply
prove that cos(π/6-a)*cos(π/3+b)-sin(π/6-a)*sin(π/3+b)=sin(a-b)
Tejas Reply
why {2kπ} union {kπ}={kπ}?
Huy Reply
why is {2kπ} union {kπ}={kπ}? when k belong to integer
Huy
if 9 sin theta + 40 cos theta = 41,prove that:41 cos theta = 41
Trilochan Reply
what is complex numbers
Ayushi Reply
Please you teach
Dua
Yes
ahmed
Thank you
Dua
give me treganamentry question
Anshuman Reply
Solve 2cos x + 3sin x = 0.5
shobana Reply
Practice Key Terms 5

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask