1. Definitions: Sum, Difference, Product, and Quotient of Functions
Let f and g be two functions. The sum of f + g, the difference f – g, the product fg, and the quotient f /g are functions whose domains are the set of all real numbers common to the domains of f and g, defined as follows:
Sum: (f + g)(x) = f (x) + g(x)
Difference: (f – g)(x) = f (x) – g(x)
Product: (f • g)(x) = f (x) • g(x)
Quotient: (f / g)(x) = f (x)/g(x), provided g(x)  0 /

2. Example: Combinations of Functions
Let f(x) = x2 – 3 and g(x)= 4x Find
(f + g) (x)
(f + g)(3)
(f – g)(x)
(f * g)(x)
(f/g)(x)
What is the domain of each combination?

3. Example: Combinations of Functions
If f(x) = 2x – 1 and g(x) = x2 + x – 2, find:
(f-g)(x)
(fg)(x)
(f/g)(x)
What is the domain of each combination?

4. Review For Test 1 Equations of Lines
Slope of a line, parallel & perpendicular
Graph with intercepts, graph using slope-intercept method
Formulae for distance and midpoint
Graph circle with standard and general form (complete the square)
Difference Quotient
Piecewise-defined functions
Domain & range of a function
Intervals where the function increases, decreases, and/or is constant
Be able to determine when a relation is a function – ordered pairs or graph
Relative Max and Min
Average Rate of change
Even and odd functions, symmetry
Transformations
Algebra of functions, domain

5. The Composition of Functions
The composition of the function f with g is denoted by f o g and is defined by the equation
(f o g)(x) = f (g(x)).
The domain of the composition function f o g is the set of all x such that
x is in the domain of g and
g(x) is in the domain of f.

6. Example: Forming Composite Functions
Given f (x) = 3x – 4 and g(x) = x2 + 6, find:
a. (f o g)(x) b. (g o f)(x)
Solution
We begin the composition of f with g. Since (f o g)(x) = f (g(x)), replace each occurrence of x in the equation for f by g(x).
f (x) = 3x – 4
This is the given equation for f.
(f o g)(x) = f (g(x)) = 3g(x) – 4
= 3(x2 + 6) – 4
= 3x – 4
= 3x2 + 14
Replace g(x) with x2 + 6.
Use the distributive property.
Simplify.
Thus, (f o g)(x) = 3x

7. Example: Forming Composite Functions
Given f (x) = 3x – 4 and g(x) = x2 + 6, find:
a. (f o g)(x) b. (g o f)(x)
Solution
b. Next, (g o f )(x), the composition of g with f. Since (g o f )(x) = g(f (x)), replace each occurrence of x in the equation for g by f (x).
g(x) = x2 + 6
This is the given equation for g.
(g o f )(x) = g(f (x)) = (f (x))2 + 6
= (3x – 4)2 + 6
= 9x2 – 24x
= 9x2 – 24x + 22
Replace f (x) with 3x – 4.
Square the binomial, 3x – 4.
Simplify.
Thus, (g o f )(x) = 9x2 – 24x Notice that (f o g)(x) is not the same as (g o f )(x).