Combinations of Functions and Composite Functions

Domain of a Function

The domain of a function is the set of all real numbers for which the function is defined. When determining the domain, exclude any values that cause division by zero or result in an even root (such as a square root) of a negative number.

• Key Point: For a function f(x), the domain is all real numbers except those that make the denominator zero or the radicand of an even root negative.

• Example: Find the domain of f(x) = \frac{1}{x-2}. Solution: Exclude x = 2 because it makes the denominator zero. Thus, the domain is all real numbers except 2.

Sum, Difference, Product, and Quotient of Functions

Functions can be combined using addition, subtraction, multiplication, and division. The domain of the resulting function is the intersection of the domains of the original functions (for the quotient, also exclude values where the denominator is zero).

• Sum:

• Difference:

• Product:

• Quotient: , where

Example: Let and .

• a.

• b. The domain of is all such that , so . The domain of is all real numbers. The domain of is .

Composition of Functions

The composition of the function f with g is denoted by and is defined as . The domain of the composite function consists of all in the domain of such that is in the domain of .

• Key Point: The domain of is the set of all such that:

  • is in the domain of

  • is in the domain of

• Example: Given and , find .

Solution:

• First, compute (domain: ).

• Then, .

• Domain: (since is defined for and is defined for all real numbers).

Excluding Values from the Domain of Composite Functions

When finding the domain of a composite function, exclude any values that are not in the domain of , as well as any for which is not in the domain of .

• Example: Given and , find .

Solution:

• First, (domain: ).

• Then, .

• Domain: (since is only defined for ).

Expressing a Function as a Composition

Some functions can be written as the composition of two simpler functions. This is useful for analyzing and understanding complex functions.

• Example: Express as a composition of two functions.

Solution:

• Let and .

• Then .

Summary Table: Operations with Functions

Additional info: When working with composite functions, always check both the inner and outer functions for domain restrictions. For example, if produces a value not allowed in , that must be excluded from the domain of the composite function.