Section 1.7
Combinations of Functions:
Composite Functions
3x+5
x2  4x  5
x2  4x  5  0
g(x)=
 x  5 x  1  0
 x  5  0  x  1  0
x=5
x=-1
Domain:  -,-1   1,5    5,  
h( x )  2 x  5
2x  5  0
2x  5
5
x
2
5
Domain :  ,  
2
Domains of Other Functions
f ( x)  4 x  5
Domain :  ,  
f ( x)  x 2  7 x
Domain  -, 
f ( x)  3 x
Domain :  ,  
Example
Find the domain of the function
4x-1
f(x)=
3x+2
Example
Find the domain of the function
f(x)= 4x-1
f
can be
g
simplified, determine the domain
before simplifying.
Example;
If the function
f(x)= x 2  4 and g(x)=x-2
f
f
x  2 in ; Domain of :  , 2    2,  
g
g
f 
x 2  4  x  2  x  2 

 x2
  x 
g
x

2
x

2


 
Determining Domains When Adding or
Subtracting Functions
The domain of f+g is the set of all real numbers that
are common to the domain of f and the domain of g.
Thus we must find the domains of f and g before
finding their intersection.
Suppose f ( x)  x+3 and g(x)= x-2 then
(f+g)(x)= x+3  x  2
Now for their domains.
f ( x)  x+3
g(x)= x-2
x3 0
x-2  0
x  3
x2
So the domain for the sum of the functions
is x  2 which in interval notation is  2, 
Continued on next slide
The graph of (f+g)(x)= x+3 
x2
confirms that the domain of this function is  2, 
y






x













Continuation of the same problem.










Determining Domains when Multiplying Functions
The domain of f  g is the set of all real numbers that
are common to the domain of f and the domain of g.
Thus we must find the domains of f and g before
finding their intersection.
5
3
Suppose f ( x) 
and g(x)=
then
x
x-2
15
(fg)(x)=
x  x-2 
y





Now for their domains.
5
3
f ( x) 
g(x)=
x
x-2
x0
x-2  0
x2
So the domain for the product of the functions
is x  0, x  2 which in interval notation is



 -,0    0, 2    2,  




















Example
If f(x)=5x-1 g(x)=x 2  2 x  1 Find each of the following:
(f+g)(x)
(f-g)(x)
(fg)(x)
f 
  ( x)
g
Example
If f(x)=5x-1 g(x)=5x 2  9 x  2 Find the domain
of the following:
(fg)(x)
f 
  ( x)
g
Example
1
1
Find the domain
g(x)=
If f(x)=
2x 1
x
of the following:
(fg)(x)
f 
  ( x)
g
Example
If f(x)= x-1 g(x)= x-6 Find the domain for:
(f-g)(x)
f(g(x))=0.85x - 300
We read this equation as "f of g of x is equal to 0.85x-300."
We call f(g(x)) the composition of the function f with g, or a
composite function. This composite function is written f g
The domain of f g is  ,0    0,3  3,  
Example
3
2
Given f(x)=
and g(x)= .
x-4
x
a. Find  f g  x  b. Find the domain of f g
Example
2
Given f(x)=
and g(x)= x.
x-3
a. Find  f g  x  b. Find the domain of f g
22
hh(( xx)) 
x
 x 
 33xx 
 22 can
can be
be written
written as
as the
the
composition
composition of
of what
what two
two functions?
functions?
22
g(x)=
g(x)=xx
ff (( xx)) 


 33xx 
 22
xx
hh(( xx)) 
  ff gg  xx 
Example
Express h(x) as a composition of two functions:
h( x )   x  6 x  5 
2
4
Example
Express h(x) as a composition of two functions:
1
h( x )  2
9 x  64
Find the domain of the function
3x-1
f(x)= 2
x  6x  7
(a)
(b)
(c)
(d)
 , 1   1,7    7,  
 ,1  1,7    7,  
 , 1  7,  
 , 1   1,7  7,  
2
If f(x)=3x-1 and g(x)=x ,
Find (f+g)(x)
(a) 3x 2  1
(b) (3 x  1) 2
(c) x 2  3x  1
(d)  x 2  3x  1
Find the domain of  f g  x 
if f(x)= x-4 and g(x)=4  3 x
(a)  0,  
(b)  3,  
(c)
(d)
 , 3
 ,0