# 3.11 Function concepts -- composite functions

 Page 1 / 1
This module describes composite functions in Algebra.

You are working in the school cafeteria, making peanut butter sandwiches for today’s lunch.

• The more classes the school has, the more children there are.
• The more children there are, the more sandwiches you have to make.
• The more sandwiches you have to make, the more pounds (lbs) of peanut butter you will use.
• The more peanut butter you use, the more money you need to budget for peanut butter.

...and so on. Each sentence in this little story is a function. Mathematically, if $c$ is the number of classes and $h$ is the number of children, then the first sentence asserts the existence of a function $h\left(c\right)$ .

The principal walks up to you at the beginning of the year and says “We’re considering expanding the school. If we expand to 70 classes, how much money do we need to budget? What if we expand to 75? How about 80?” For each of these numbers, you have to calculate each number from the previous one, until you find the final budget number.

 $\underset{\to }{\text{# Classes}}$ $\underset{\to }{\text{# Children}}$ $\underset{\to }{\text{# Sandwiches}}$ $\underset{\to }{\text{lb.}}$ $\underset{\to }{\text{}}$

But going through this process each time is tedious. What you want is one function that puts the entire chain together: “You tell me the number of classes, and I will tell you the budget.”

 $\underset{\to }{\text{# Classes}}$ $\underset{\to }{\text{}}$

This is a composite function —a function that represents in one function, the results of an entire chain of dependent functions . Since such chains are very common in real life, finding composite functions is a very important skill.

## How do you make a composite function?

We can consider how to build composite functions into the function game that we played on the first day. Suppose Susan takes any number you give her, quadruples it, and adds 6. Al takes any number you give him and divides it by 2. Mathematically, we can represent the two functions like this:

$S\left(x\right)=4x+6$
$A\left(x\right)=\frac{x}{2}$

To create a chain like the one above, we give a number to Susan; she acts on it, and gives the resulting number to Al; and he then acts on it and hands back a third number.

$3\to \text{Susan}\to S\left(3\right)=18\to \text{Al}\to A\left(18\right)=9$

In this example, we are plugging $S\left(3\right)$ —in other words, 18— into Al’s function. In general, for any $x$ that comes in, we are plugging $S\left(x\right)$ into $A\left(x\right)$ . So we could represent the entire process as $A\left(S\left(x\right)\right)$ . This notation for composite functions is really nothing new: it means that you are plugging $S\left(x\right)$ into the $A$ function.

But in this case, recall that $S\left(x\right)=4x+6$ . So we can write:

$A\left(S\left(x\right)\right)=\frac{S\left(x\right)}{2}=\frac{4x+6}{2}=2x+3$

What happened? We’ve just discovered a shortcut for the entire process. When you perform the operation $A\left(S\left(x\right)\right)$ —that is, when you perform the Al function on the result of the Susan function—you are, in effect, doubling and adding 3. For instance, we saw earlier that when we started with a 3, we ended with a 9. Our composite function does this in one step:

$3\to 2x+3\to 9$

Understanding the meaning of composite functions requires real thought. It requires understanding the idea that this variable depends on that variable, which in turn depends on the other variable; and how that idea is translated into mathematics. Finding composite functions, on the other hand, is a purely mechanical process—it requires practice, but no creativity. Whenever you are asked for $f\left(g\left(x\right)\right)$ , just plug the $g\left(x\right)$ function into the $f\left(x\right)$ function and then simplify.

## Building and testing a composite function

$f\left(x\right)={x}^{2}-4x$

$g\left(x\right)=x+2$

What is $f\left(g\left(x\right)\right)$ ?

• To find the composite, plug $g\left(x\right)$ into $f\left(x\right)$ , just as you would with any number.

$f\left(g\left(x\right)\right)=\left(x+2{\right)}^{2}-4\left(x+2\right)$

• Then simplify.

$f\left(g\left(x\right)\right)=\left(x{}^{2}\text{}+4x+4\right)-\left(4x+8\right)$

$f\left(g\left(x\right)\right)={x}^{2}-4$

• Let’s test it. $f\left(g\left(x\right)\right)$ means do $g$ , then $f$ . What happens if we start with $x=9$ ?

$7\to g\left(x\right)\to 7+2=9\to f\left(x\right)\to \left(9{\right)}^{2}-4\left(9\right)=45$

• So, if it worked, our composite function should do all of that in one step.

$7\to {x}^{2}-4=\left(7{\right)}^{2}-4=45$ $✓$ It worked!

There is a different notation that is sometimes used for composite functions. This book will consistently use $f\left(g\left(x\right)\right)$ which very naturally conveys the idea of “plugging $g\left(x\right)$ into $f\left(x\right)$ .” However, you will sometimes see the same thing written as $f°g\left(x\right)$ , which more naturally conveys the idea of “doing one function, and then the other, in sequence.” The two notations mean the same thing.

how can chip be made from sand
are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!