Algebra 2
Name___________________________________
Operations on Functions
Operations on Functions
Operation
Addition
Subtraction
Multiplication
Division
Definition
Example
Let f(x)=2x and
g(x)=-x+5
(f+g)(x)=f(x)+g(x)
(f-g)(x)=f(x)-g(x)
(f  g )(x)= f (x)  g (x)
2x+(-x+5)=x+5
2x-(-x+5)=3x - 5
2x(-x+5)=-2x2+10x
f
f ( x)
 ( x) 
, g ( x)  0
g ( x)
g
2x
,x  5
 x5
Example 1: Given f(x) =x2 - 4 and g(x) =2x+1, find each function. Indicate any restrictions in
the domain or range.
a. (f+g)(x)
b. (f-g)(x)
You Try: Given f(x)=x2+5x-2 and g(x)=3x-2, find each function.
a. (f+g)(x)
b. (f-g)(x)
Given f(x) =3x2+7x and g(x) =2x2-x-1, find each function. Indicate any restrictions in the
domain.
a. (f+g)(x)
b. (f-g)(x)
In Example 1, the functions f(x) and g(x) have the same domain of all real numbers. The
functions (f+g)(x) and (f-g)(x) also have domains that include all real numbers. Under division,
the domain of the new function is restricted by excluded values that cause the denominator to
equal zero.
Example 2: Given f(x) =x2+7x+12 and g(x) =3x-4, find each function. Indicate any restrictions
in the domain or range.
a. ( f  g )( x)
f
b.  (x)
g


You Try: Given f(x) =x2-7x+2 and g(x) = x+4, find each function.
a. ( f  g )( x)
f
 (x)
b.  g 
Given f(x) = 3x2 - 2x+1 and g(x) =x-4, find each function. Indicate any restrictions in the
domain and range.
a. ( f  g )( x)
f
 (x)
b.  g 
Composition of Functions:
Another method used to combine functions is a composition of functions. In a composition of
functions, the results of one function are used to evaluate a second function.
Suppose f and g are functions such that the range of g is a subset of the domain of f.
Then the composition of function f  g can be described by
 f  g ( x)  f g ( x)
The composition of two functions may not exist. Given two functions f and g,  f  g (x) is
defined only if the range of g(x) is a subset of the domain of f. Likewise, g  f (x) is defined
only if the range of f(x) is a subset of the domain of g.
Example 3: Find  f  g (x)
a) f(x) =2a-5, g(x)=4a
You Try: Find  f  g (x)
a) f(x)=x2+2 and g(x)=x-6
Example 4: Find g  f (x)
a) f(x) =2a-5, g(x)=4a
You Try: Find g  f (x)
b) f(x)=x2+2 and g(x)=x-6
Example 5: Find f g (3)
a) f(x) =2a-5, g(x)=4a
You Try: Find f ( g (1))
b) f(x)=x2+2 and g(x)=x-6