1. Pre-Calc Lesson 4.2
Operations on Functions
We can do some basic operations on functions.
We can add, subtract, multiply and divide functions.
If given functions f(x) and g(x), then:
Sum of functions f and g : (f + g)(x) = f(x) + g(x)
Difference of functions f and g: (f - g)(x) = f(x) – g(x)
Product of functions f and g: (f g)(x) = f(x) g(x)
Quotient of functions f and g: (f/g)(x) = f(x)/g(x) where g(x) ≠ 0

2. Let f(x) = x + 1 and g(x) = x2 – 1, find a ‘rule’ for each of
Example 1:
Let f(x) = x + 1 and g(x) = x2 – 1, find a ‘rule’ for each of
the following functions.
(f + g)(x)  think this means f(x) g(x)
 (x+1) (x2 – 1)
(drop all parentheses)  x x2 - 1
(rearrange terms)  x2 + x (voila!)
(f/g)(x) think  f(x)
g(x)
replace  x + 1
x2 – 1
reduce if possible ??  (x + 1)
factor  (x+1)(x-1)
Cancel like factors 
(x – 1)
ta da!  but we must state: x ≠ know why ?????

3. Another way of combining functions is called:
Composition of functions!
This is simply a process of substituting a functions ‘rule’
In for the variable in a 2nd function.
Example 2:
Let f(x) = x4 – 3x2 and g(x) = √(x – 2)
Find the composition of functions f and g.
This means  (f of g)(x)
this is what it looks like f(g(x))
and this means  Substitute the rule from g(x) in to the variable in the rule of f(x) ???
i.e. Plug √(x – 2) in for ‘x’ in the rule (x)4 - 3(x)2
 (√(x – 2))4 – 3(√(x – 2))2
 ( √(x – 2)2)2 - 3(x – 2)
 (x – 2)2 – 3x + 6
 x2 – 4x + 4 – 3x + 6
 x2 – 7x for which x > 2 ?? Voila!

4. Let f(x) = 1/x and g(x) = x + 1, find new ‘rules’ for f(g(x))
Example 3:
Let f(x) = 1/x and g(x) = x + 1, find new ‘rules’ for f(g(x))
and g(f(x)) and give the domain of each new ‘composite’
function.
f(g(x)) = f(x + 1) 1
(x + 1) ; x ≠ - 1
g(f(x)) = g( 1 ) x
( 1 ) + 1
and x ≠ 0
Hw: pgs : CE: #1-13 all,
WE: #1-19 odd