Evaluating Functions

Understanding Function Notation

A function is a rule that assigns to each input exactly one output. The notation f(x) represents the value of the function f at the input x. Evaluating a function means substituting a specific value or expression for x and simplifying.

• Function Notation: If f(x) = x^2 - 5x - 9, then f(5) means substitute 5 for x in the formula.

• General Rule: To evaluate f(a), replace every x in the function with a.

Examples of Function Evaluation

• Example 1: Evaluating at a Number Given f(x) = x^2 - 5x - 9, find f(5).

  • Substitute 5 for x:

• Example 2: Evaluating at an Expression Find f(x+3).

  • Substitute (x+3) for x:

  • Expand using the formula :

  • So,

  • Combine like terms:

• Example 3: Evaluating at a Negative Expression Find f(-2x).

  • Substitute -2x for x:

Key Points

• Substitution: Always replace every occurrence of x with the new input (number or expression).

• Order of Operations: Follow the correct order: parentheses, exponents, multiplication/division, addition/subtraction.

• Expanding Expressions: When substituting an expression, use algebraic expansion formulas as needed.

Summary Table: Function Evaluation Examples

Additional info:

• When evaluating functions at expressions, always expand and simplify fully to standard form.

• These skills are foundational for later topics such as graphing, transformations, and composition of functions.