Overview

This topic introduces functions, notation, evaluation, and interpretation. Students will understand how to identify functions, use domain and range, and apply function notation to solve real-world problems. Recognizing the structure and purpose of a function is a key algebraic skill.

Key Concepts and Structures

• Function Definition: A rule that assigns each input exactly one output. Common notation: f(x).
• Domain: The set of all possible input values (x-values) for which the function is defined.
• Range: The set of possible output values (f(x)-values) the function produces.
• Function Notation: f(x) is read as “f of x.” It replaces y in equations and helps describe outputs for specific inputs.
• Evaluating Functions: Replace x in the expression with the given value.
  Example: If f(x) = 2x + 3, then f(4) = 2(4) + 3 = 11.
• Piecewise Functions: Defined by different expressions for different intervals of x. Pay attention to conditions on each piece.
• Function Operations: Add, subtract, multiply, and divide functions. Also includes composition: (f ◦ g)(x) = f(g(x)).
• Vertical Line Test: Used to determine if a graph represents a function. A vertical line must intersect the graph at most once.

Step-by-Step Example

Problem: Evaluate f(x) = 3x^2 - x + 5 at x = -2

Step 1: Plug -2 into the function:
f(-2) = 3(-2)^2 - (-2) + 5

Step 2: Simplify:
f(-2) = 3(4) + 2 + 5 = 12 + 2 + 5 = 19

Final Answer: f(-2) = 19

Quick Tip