Composite functions
[Type the document subtitle]
Composite functions, working them out.
luxvis
11/19/2012
Composite Functions
What are they?
In the real world, it is not uncommon for the output of one thing to depend on the input of another function. For
example the amount of tax we would pay depends on the gross salary the person makes. Such functions are called
composite functions.
So a function is performed first and then a second function is performed on the result of the first function, that is
what is actually taking place when we composition.
Special terminology
The composite function f
g , the composition of f and g is defined as follows
( f g )( x)  f ( g ( x))
For the above function to be defined or to exist then a certain condition must be met namely
Range g ( x)  domain f ( x)
Formula
What f
Definitions
g it means
For f
g to be defined
What is the domain of f g
What is the range of f g
f g
( f g )( x)  f ( g ( x))
Range g ( x)  domain f ( x)
domain f g  domain f
range f g  range g
Example- find f g
Let us consider an example and we will see how this works in practice
Consider the following two functions
f ( x)  x 2 and g ( x)  x  4
Say we will like to find the following composite function f g and whether this function exists
Step 1- sketch both functions to see what they look like and determine their domains and ranges
Sketch the two functions and find their respective domains and ranges
f ( x)  x 2
Equation of the graph
f ( x)  x 2
Domain of the graph- what is the values of x it can
have
Range of the graph- values of y it can have
 ,  
[0, )
Now to sketch the graph of g ( x)  x  4
g ( x)  x  4
Equation of the graph
 ,  
Domain of the graph- what is the values of x it can
have
Range of the graph- values of y it can have
 ,  
Now the reason we plotted these two graphs is to help us understand the restrictions that must be placed on the
domain for the various composite functions to be defined
Step-2- Work out f
g
( f g )( x)  f ( g ( x))
And this gives us the following
Find f
g
f g ( x)  f ( g ( x))
 f ( x  4)
 ( x  4) 2
Step -3- Work out if the function is defined
Now for this function to be defined the condition Range g ( x)  domain f ( x)
This means that the range is within the domain of the f ( x)
Let’s see how this looks
We have the following operation x  g ( x)  f ( x)
Domain g(x)
Range g(x)
Domain : R - (-∞, ∞)
Range: R - (-∞, ∞)
Domain f(x)
Range f(x)
Domain: R - (-∞, ∞)
Range: R- (0, ∞)
This is a way of showing the step visually. Notice how I am using the number line to my advantage!
So since range of g ( x) is a subset of the domain of f ( x) then this composite function exists. I like to put the
domain of f ( x) on the bottom and the range of g ( x) on the top , as it helps me see if the range is within the
domain of f ( x)
here are the answers
f g
What f
( f g )( x)  f ( g ( x))   x  4
g it means
2
Range g ( x)  domain f ( x)
For f
g to be defined
What is the domain of f g
domain f g  domain f which is  ,  
What is the range of f
range f g  range g which is  ,  
g
Example-2
Now let’s see if we can find the composite function g
Let’s follow the previous steps
f
g f ( x)  g ( f ( x))
 g ( x2 )
 x2  4
For this to be defined then Range f ( x)  domain g ( x)
Now this is the most important step as it shows the actual process that is taking place here
x  f ( x)  g ( x )
So that gives us the following
Domain f(x)
Range f(x)
[0, ∞)
Domain g(x)
(-∞,∞)
Range g(x)
Now remember that
Range f ( x)  domain g ( x)
The range of f(x) is a subset of the domain of g(x), so this composite function is defined.
Important Points to Remember
g f
Formula- g f
What g f it means
For g f to be defined
What is the domain of g f
What is the range of g f
Definitions
( g f )( x)  g ( f ( x))
Range f ( x)  domain g ( x)
domain g f  domain g
range g f  range f
Difficult Example
Consider the following two functions f :{x : x  3}  R, f ( x)  3  x and g : R  R, g ( x)  x 2  1
a) Show that f g is not defined
Let us sketch both graphs and work out their domains and ranges before we answer the question
f :{x : x  3}  R, f ( x)  3  x
The graph that is been plotted
f :{x : x  3}  R, f ( x)  3  x
Domain of the graph
Range of the graph
 ,3
0,  
The graph that is been plotted
g : R  R, g ( x)  x 2  1
Domain of the graph
 ,  
 1,  
Range of the graph
Now for f g to be defined the condition g : R  R, g ( x)  x 2  1
Let’s see how this looks
We have the following operation x  g ( x)  f ( x)
Domain g(x)
Range g(x)
Domain f(x)
Range f(x)
So it is clear that range g(x) is not a subset of the domain f(x). So the composite function f g is not defined. Of
course we could define it if we restrict the domain of f to [-1, 3]. Remember domain f g  domain f
Composite Functions
What does f g ( x)
1
mean?
2
How do we know if f °
g exists?
It means we substitute function
g(x) into f(x)
For it to exist or be defined then it
must meet this condition:
g : R  R, g ( x)  x 2  1
f :{x : x  3}  R, f ( x)  3  x
Let say f(x) = 3x and g(x) = x+2
Find f ° g
Answer
f :{x : x  3}  R, f ( x)  3  x
One way is to find the domain for
each function and then plot the
range of g(X) and the domain of
f(x) and see if the condition meets
3
=f (x+2) = 3(x+2) = 3x+6
Now is this defined?
g : R  R, g ( x)  x 2  1
First we find domain and range of
each of the above function
range of g(x) = R = (-∞,∞)
domain of f(x) = R = (-∞,∞)
What is the best way of
doing these type of
questions
Draw a number line and you will see
that range is a subset or equal to the
domain of f(x)
Range g(x): (-∞,
∞)
0
Domain f(x) :(-∞,
∞)
4
5
What happens if it is
g f ( x)
What is the definition
for
f g ( x)
Everything is done the same except
this time the definition
For g f
domain g f  domain g
range g f  range f
For f g
domain f g  domain f
range f g  range g