Functions are the building blocks of calculus, defining relationships between inputs and outputs. They're essential for modeling real-world scenarios and form the foundation for more advanced mathematical concepts.

Understanding function basics, notation, and classifications is crucial for mastering calculus. These concepts help us analyze and manipulate functions, setting the stage for exploring limits, derivatives, and integrals in future lessons.

Function Basics and Notation

Domain and range of functions

• A function relates a set of inputs (domain) to a set of outputs (range) where each input corresponds to exactly one output
• Domain consists of all possible input values for a function, usually represented by the variable $x$
• Range consists of all possible output values for a function, usually represented by the variable $y$
• Identify the domain and range by considering:
  • Restrictions on input values (division by zero, square roots of negative numbers)
  • Output values resulting from the function's equation

Function notation and evaluation

• Function notation: $f(x)$ or $y = f(x)$
  • $f$ denotes the function name
  • $x$ denotes the input variable
  • $f(x)$ or $y$ denotes the output value
• Evaluate a function for a given input by substituting the input value for the variable in the function's equation and simplifying
  • If $f(x) = 2x + 1$, then $f(3) = 2(3) + 1 = 7$
• Function composition combines two or more functions to create a new function
  • Notation: $(f \circ g)(x) = f(g(x))$
  • Evaluate a composite function by first evaluating the inner function, then using that result as the input for the outer function

Vertical line test for functions

• A relation is a function if and only if every vertical line intersects the graph of the relation at most once
• Vertical line test determines if a relation is a function based on its graph
  • If any vertical line intersects the graph more than once, the relation is not a function
  • If no vertical line intersects the graph more than once, the relation is a function

Classifications of functions

• One-to-one (injective) functions
  • Each element in the codomain pairs with at most one element in the domain
  • No two distinct inputs map to the same output
  • Horizontal line test: If any horizontal line intersects the graph more than once, the function is not one-to-one
• Onto (surjective) functions
  • Each element in the codomain pairs with at least one element in the domain
  • Every possible output value is achieved by at least one input value
• Bijective functions
  • A function that is both one-to-one and onto
  • Each element in the codomain pairs with exactly one element in the domain
  • Bijective functions have an inverse function