1. Lesson 51
Operations & Composition Functions
Text: Chapter 3, section 9
 A function is a relationship between x and y in which each x-value has
exactly one y-value.
 X-values are the input or the domain of the function and the y-values are
outputs or range of the function.
 Functions are written this way:
f ( x)  x 2  1
If f ( x)  x 2  1 , then f (2)  (2) 2  1
5
This is the point (2, 5). Any x-value from the domain will result in another
point.
 REMEMBER – the vertical line test to check if a graph is a function.
 Operations of Functions:
( f  g )( x)  f ( x)  g ( x)
( f  g )( x)  f ( x)  g ( x)
( f  g )( x)  f ( x) g ( x)
 f  f ( x)
   x  , g ( x)  0
g g ( x)
 The domains of these functions are restricted to the intersection of the
domains of f ( x) and g ( x) .

2. Example:
If f ( x)  x 2  5 x  2
g ( x)  3x  4
Find:
A) ( f  g )( x)
B) ( f  g )( x)
C) ( f  g )( x)
Solution:
A) ( f  g )( x)  f ( x)  g ( x)
 ( x 2  5 x  2)  (3x  4)
 x2  8x  6
B) ( f  g )( x)  f ( x)  g ( x)
 ( x 2  5 x  2)  (3x  4)
 x2  2 x  2
C) ( f  g )( x)  f ( x) g ( x)
 ( x 2  5 x  2)(3x  4)
 3x 3  19 x 2  26 x  8
Composition of a function:
 The composition of a function f and g, is written as ( f  g )( x) and is defined
as f ( g ( x)) .
 The function g is substituted into the function f.
Example:
If f ( x)  x  1 and g ( x)  x  2 find f ( g (2))
g (2)  (2)  2  f (4)  (4)  1
=4 =3
f ( g ( x))  f ( x  2)
 ( x  2)  1
 x 1

3. ( g  f )( x)  g ( f ( x))
 g ( x  1)
 ( x  1)  2
 x 1
Usually ( f  g )( x)  ( g  f )( x)
Assignment: Exercise 51 Q 1 to 17