Functions

Definition of a Function

A function is a relation that assigns exactly one output value for each input value from a given set. Functions are commonly denoted as f(x), g(x), etc., where x is the input variable.

• Domain: The set of all possible input values (x-values) for which the function is defined.

• Range: The set of all possible output values (f(x)-values) produced by the function.

Given Functions

Function Evaluation

Evaluating Functions

To evaluate a function at a specific value, substitute the given value for the variable x in the function's formula and simplify.

• Example: To evaluate for :

  • Substitute into the function:

  • Simplify:

Composite Functions

A composite function is formed when one function is evaluated inside another. The notation means .

• Example:

• First, evaluate , then use that result as the input for .

Sample Problem

Evaluate the number .

1. First, evaluate :

2. Next, use this result as the input for :

   • Expand:

   • Combine like terms:

Final Answer:

Summary Table: Function Evaluation Steps