<< 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

difference between calculus and pre calculus?
Asma Reply
give me an example of a problem so that I can practice answering
Jenefa Reply
dont forget the cube in each variable ;)
of she solves that, well ... then she has a lot of computational force under her command ....
what is a function?
CJ Reply
I want to learn about the law of exponent
Quera Reply
explain this
Hinderson Reply
what is functions?
Angel Reply
A mathematical relation such that every input has only one out.
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
If the plane intersects the cone (either above or below) horizontally, what figure will be created?
Feemark Reply
can you not take the square root of a negative number
Sharon Reply
No because a negative times a negative is a positive. No matter what you do you can never multiply the same number by itself and end with a negative
Actually you can. you get what's called an Imaginary number denoted by i which is represented on the complex plane. The reply above would be correct if we were still confined to the "real" number line.
Suppose P= {-3,1,3} Q={-3,-2-1} and R= {-2,2,3}.what is the intersection
Elaine Reply
can I get some pretty basic questions
Ama Reply
In what way does set notation relate to function notation
is precalculus needed to take caculus
Amara Reply
It depends on what you already know. Just test yourself with some precalculus questions. If you find them easy, you're good to go.
the solution doesn't seem right for this problem
Mars Reply
what is the domain of f(x)=x-4/x^2-2x-15 then
Conney Reply
x is different from -5&3
All real x except 5 and - 3
how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
jeric Reply
Don't think that you can.
By using some imaginary no.
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
jeric Reply
What are the question marks for?
Practice Key Terms 5

Get the best Precalculus course in your pocket!

Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

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