Acc Math 2
Today’s Question:
How do you perform operations with
functions?
Standard: MM2A5.d
Lesson 4.2
Operations with Functions
Objective
 To define the sum, difference, product, and quotient
of functions.
 To form and evaluate composite functions.
 To determine the domain for composite functions.
Basic function operations
 Sum
 f + g  x   f  x  + g  x 
 Difference
 f – g  x   f  x  – g  x 
 Product
f
 Quotient
f x
f 
, g x  0
 f g  x      x  
g x
g 
g  x   f  x  g  x 
Function, domain, & range
 The domain of a function is the set of all input values
(x-values)
 The range of a function is the set of all output values
(y values)
 A relation is a function if all values of the domain are
unique (they do not repeat).
 A test to see if a relation is a function is the vertical
line test.
 If it is possible to draw a vertical line and cross the graph
of a relation in more than one point, the relation is not
a function.
Example 1
 Find each function and state its domain:
f  x   4 x 5 ; g  x   2x 5
 f+g
 f  g  x   4 x 5  2 x 5  6 x 5 ; Df g   x : All Re al 
 f–g
 f  g  x   4 x 5  2 x 5  2 x 5 ; Df g   x : All Re al 
 f ·g
 f  g  x   4 x 5  2 x 5  8 x 10 ; Df g   x : All Re al 
 f /g
 f / g  x   4 x 5 / 2x 5  2; Df g  x : x  0
Example 2
 Find each function and state its domain:
f  x   2x 2 ; g  x   x 4
 f ·g
 f  g  x    2 x 2  x 4   2 x 6 ; Df g   x : All Re al 
 f /g
2x 2
2
 f g  x   4  2 ; Df
x
x
g
  x : x  0
Your Turn
 Find each function and state its domain:
f  x   6 x 2 ; g  x   3 x 2
 f+g
 f–g
 f ·g
 f /g
Composition of functions
 Composition of functions is the successive
application of the functions in a specific order.
 Given two functions f and g, the composite function
f g is defined by  f g  x   f  g  x   and is read
“f of g of x.”
 The domain of f g is the set of elements x in the
domain of g such that g(x) is in the domain of f.
 Another way to say that is to say that “the range of function
g must be in the domain of function f.”
Example 1
 Evaluate  f g  x  and  g f  x  :
f
x  x  3
g
 x   2x
2


gf  fgxx  2 2
 xx 2 31  31
1
2


 22 xx2246 x  9  1
 2 x 2  12 x  18  1
f
g  x   2 x 2  4
 g f  x   2x 2  12x  17
You can see that function composition is not commutative!
Example 2
 Evaluate  f g  x  and  g f  x  :

f  x   2x

g x  x
3
1
      2x
fg gf x x  2x
1 3
 2 x 3
2
 3
x
2
 f g  x   3
x
1
 g f  x   3
2x

3 1
1
3
2x
Again, not the same function. What is the domain???
Example 3
 Find the domain of  f g  x  and  g f  x :

f x  x 1
g
x 
f
x
g  x   x  1
Df
g
 x : x  0
(Since a radicand can’t be negative in the set of real numbers,
f be
x  greater
 x than
1 or
Dgequal
x  1
xgmust
f   xto: zero.)
(Since a radicand can’t be negative in the set of real numbers,
x – 1 must be greater than or equal to zero.)
Your turn
 Evaluate  f g  x  and  g f  x  :

f  x   3x 2

g x  x  5
Example 4
 Find the indicated values for the following functions
if:

f  x   2x  3
2
g
x

x
1
 

f (g (1))
f (g (4))
g (f (2))
g (g (2))
Summary…
 Function arithmetic – add the functions (subtract, etc)
 Addition
 Subtraction
 Multiplication
 Division
 Function composition
 Perform function in innermost parentheses first
 Domain of “main” function must include range of “inner”
function