Overview

This topic introduces the definition of a function, domain and range, function notation, and operations with functions including evaluation, composition, and inverses. A strong grasp of these concepts is essential for advanced topics in precalculus and calculus.

Key Concepts and Terms

• Function: A relation where each input has exactly one output. Often written as f(x).
• Domain: The set of all possible input values (x-values).
• Range: The set of all possible output values (f(x)-values).
• Function Notation: f(x) means the value of function f at input x.
• Evaluating Functions: Substitute the input value into the function and simplify.
• Operations: You can add, subtract, multiply, and divide functions:
  • (f + g)(x) = f(x) + g(x)
  • (f \cdot g)(x) = f(x) \cdot g(x)
• Composition of Functions: (f \circ g)(x) = f(g(x)), plug one function into another.
• Inverse Functions: A function that "reverses" another: f(f^{-1}(x)) = x. Found by solving y = f(x) for x and switching variables.

Step-by-Step Example

Problem: Given f(x) = 2x + 3 and g(x) = x^2, find (f \circ g)(2)

Step 1: Compute g(2):
g(2) = 2^2 = 4

Step 2: Plug that into f:
f(4) = 2(4) + 3 = 11

Answer: (f \circ g)(2) = 11

Quick Tip