<< Chapter < Page Chapter >> Page >

Rational function

A rational function    is a function that can be written as the quotient of two polynomial functions P ( x )   and   Q ( x ) .

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

Solving an applied problem involving a rational function

A large mixing tank currently contains 100 gallons of water into which 5 pounds of sugar have been mixed. A tap will open pouring 10 gallons per minute of water into the tank at the same time sugar is poured into the tank at a rate of 1 pound per minute. Find the concentration (pounds per gallon) of sugar in the tank after 12 minutes. Is that a greater concentration than at the beginning?

Let t be the number of minutes since the tap opened. Since the water increases at 10 gallons per minute, and the sugar increases at 1 pound per minute, these are constant rates of change. This tells us the amount of water in the tank is changing linearly, as is the amount of sugar in the tank. We can write an equation independently for each:

water:  W ( t ) = 100 + 10 t  in gallons sugar:  S ( t ) = 5 + 1 t  in pounds

The concentration, C , will be the ratio of pounds of sugar to gallons of water

C ( t ) = 5 + t 100 + 10 t

The concentration after 12 minutes is given by evaluating C ( t ) at t =   12.

C ( 12 ) = 5 + 12 100 + 10 ( 12 ) = 17 220

This means the concentration is 17 pounds of sugar to 220 gallons of water.

At the beginning, the concentration is

C ( 0 ) = 5 + 0 100 + 10 ( 0 ) = 1 20

Since 17 220 0.08 > 1 20 = 0.05 , the concentration is greater after 12 minutes than at the beginning.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

There are 1,200 freshmen and 1,500 sophomores at a prep rally at noon. After 12 p.m., 20 freshmen arrive at the rally every five minutes while 15 sophomores leave the rally. Find the ratio of freshmen to sophomores at 1 p.m.

12 11

Got questions? Get instant answers now!

Finding the domains of rational functions

A vertical asymptote    represents a value at which a rational function is undefined, so that value is not in the domain of the function. A reciprocal function cannot have values in its domain that cause the denominator to equal zero. In general, to find the domain of a rational function, we need to determine which inputs would cause division by zero.

Domain of a rational function

The domain of a rational function includes all real numbers except those that cause the denominator to equal zero.

Given a rational function, find the domain.

  1. Set the denominator equal to zero.
  2. Solve to find the x -values that cause the denominator to equal zero.
  3. The domain is all real numbers except those found in Step 2.

Finding the domain of a rational function

Find the domain of f ( x ) = x + 3 x 2 9 .

Begin by setting the denominator equal to zero and solving.

x 2 9 = 0 x 2 = 9 x = ±3

The denominator is equal to zero when x = ± 3. The domain of the function is all real numbers except x = ± 3.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Find the domain of f ( x ) = 4 x 5 ( x 1 ) ( x 5 ) .

The domain is all real numbers except x = 1 and x = 5.

Got questions? Get instant answers now!

Identifying vertical asymptotes of rational functions

By looking at the graph of a rational function, we can investigate its local behavior and easily see whether there are asymptotes. We may even be able to approximate their location. Even without the graph, however, we can still determine whether a given rational function has any asymptotes, and calculate their location.

Questions & Answers

what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
give me the waec 2019 questions
Aaron Reply
the polar co-ordinate of the point (-1, -1)
Sumit Reply
prove the identites sin x ( 1+ tan x )+ cos x ( 1+ cot x )= sec x + cosec x
Rockstar Reply
tanh`(x-iy) =A+iB, find A and B
Pankaj Reply
what is the addition of 101011 with 101010
Branded Reply
If those numbers are binary, it's 1010101. If they are base 10, it's 202021.
extra power 4 minus 5 x cube + 7 x square minus 5 x + 1 equal to zero
archana Reply
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
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
Find that number sum and product of all the divisors of 360
jancy Reply
exponential series
prime number?
what is subgroup
Purshotam Reply
Prove that: (2cos&+1)(2cos&-1)(2cos2&-1)=2cos4&+1
Macmillan 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?