1. Home
  2. Algebra and trigonometry
  3. Polynomial and rational functions
  4. Rational functions

5.6 Rational functions  (Page 3/16)

<< Chapter < Page
  Algebra and trigonometry     Page 3 / 16
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

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
what is titration
John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
hello friend how are you
Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
<< Chapter < Page Page > Chapter >>
Practice Key Terms 5

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




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