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Section 2.6Combinations of Functions:
Composite Functions
The Domain of a Function
2
2
3x+5g(x)=
x 4 5
x 4 5 0
5 1 0
5 0 1 0
x=5 x=-1
: - ,-1 1,5 5,
x
x
x x
x x
Domain
( ) 2 5
2 5 0
2x 5
5 x
25
: ,2
h x x
x
Domain
2( ) 7
- ,
f x x x
Domain
Domains of Other Functions
( ) 4 5
: ,
f x x
Domain
3( )
: ,
f x x
Domain
Example
Find the domain of the function
4x-1f(x)=
3x+2
Example
Find the domain of the function
f(x)= 4x-1
The Algebra of Functions
2
2
fIf the function can be
g
simplified, determine the domain
before simplifying.
Example;
f(x)= x 4 and g(x)=x-2
f fx 2 in ; Domain of : ,2 2,
g g
2 2f 42
g 2 2
x xxx x
x x
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 ( ) x+3 af x nd g(x)= x-2 then
(f+g)(x)= x+3 2
Now for their domains.
( ) x+3 g(x)= x-2
3 0 x-2 0
3 x 2
So the domain for the sum of the functions
is x 2 which in inter
x
f x
x
x
val notation is 2,Continued on next slide
Determining Domains When Adding or Subtracting Functions
Continuation of the same problem.
x
y
The graph of (f+g)(x)= x+3 2
confirms that the domain of this function is 2,
x
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.
5Suppose ( ) an
xf x
3d g(x)= then
x-215
(fg)(x)= x x-2
Now for their domains.
5 3( ) g(x)=
x x-20 x-2 0
x 2
So the domain for the product of the functions
is x 0, x 2 which in
f x
x
interval notation is
- ,0 0,2 2,
x
y
Determining Domains when Multiplying Functions
Example2If f(x)=5x-1 g(x)=x 2 1 Find each of the following:
(f+g)(x)
(f-g)(x)
(fg)(x)
f( )
g
x
x
Example
2If f(x)=5x-1 g(x)=5x 9 2 Find the domain
of the following:
(fg)(x)
f( )
g
x
x
Example1 1
If f(x)= g(x)= Find the domain 2 1
of the following:
(fg)(x)
f( )
g
x x
x
Example
If f(x)= x-1 g(x)= x-6 Find the domain for:
(f-g)(x)
Composite Functions
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 2Given f(x)= and g(x)= .
x-4 xa. Find f g b. Find the domain of f gx
Example
2Given f(x)= and g(x)= x.
x-3a. Find f g b. Find the domain of f gx
Decomposing Functions
Example
42
Express h(x) as a composition of two functions:
( ) 6 5 h x x x
Example
2
Express h(x) as a composition of two functions:
1( )
9 64h x
x
(a)
(b)
(c)
(d)
2
Find the domain of the function
3x-1f(x)=
x 6 7x
, 1 1,7 7,
,1 1,7 7,
, 1 7,
, 1 1,7 7,
(a)
(b)
(c)
(d)
2 If f(x)=3x-1 and g(x)=x ,
Find (f+g)(x)
2
2
2
2
3 1
(3 1)
3 1
3 1
x
x
x x
x x
(a)
(b)
(c)
(d)
Find the domain of f g
if f(x)= x-4 and g(x)=4 3
x
x
0,
3,
, 3
,0
