Chapter 1.4 - Combinations of Functions

Arithmetic Combinations of Functions

Just like two real numbers, functions can be combined by the operations of addition, subtraction, multiplication, and division.

Definitions:

The domain of an arithmetic combination of functions f and g consists of all real numbers that are common to the domains of f and g.

Examples:

If and , find the sum, difference, product, and quotient of f and g.

Sum:

Difference:

Product:

Quotient:

Compositions of Functions

Another way of combining two functions is to form the composition of one with the other.

Definition:

The composition of the function f with g is .

The domain of is the set of all x in the domain of g such that g(x) is in the domain of f.

Example:

Find for and . If possible, find and .

The domain of is [1, ∞). So = is defined, but is not defined because 0 is not in the domain of .