Functions are the building blocks of mathematics, defining relationships between inputs and outputs. This section explores the crucial concepts of domain and range, which determine the valid inputs and resulting outputs for a function.

We'll examine how to find domains for various function types, including those with algebraic restrictions or real-world constraints. We'll also dive into piecewise functions and learn different notations for expressing domains and ranges clearly.

Domain and Range of Functions

Domain of equation-defined functions

• Represents the set of all possible input values (x-values) for a function determined by considering the function's equation and any restrictions on the input values
• Algebraic restrictions on the domain arise when division by zero is undefined, so the denominator cannot equal zero
  • For rational functions in the form $\frac{f(x)}{g(x)}$, solve $g(x) \neq 0$ to find excluded x-values (vertical asymptotes)
• Even-indexed roots (square root, fourth root, etc.) are only defined for non-negative values
  • For functions with even-indexed roots, such as $\sqrt{f(x)}$, solve $f(x) \geq 0$ to find the domain (radicand must be non-negative)
• Real-world context can further limit the domain
  • Physical quantities (length, time, etc.) are typically non-negative (cannot have negative length or time)
  • Discrete quantities (number of people, items, etc.) are typically whole numbers (cannot have fractional people or items)

Piecewise functions and their properties

• Defined by different equations over different parts of the domain with the domain divided into intervals, each having its own function definition
• To graph a piecewise function:
  1. Identify the intervals of the domain and their corresponding function definitions
  2. Graph each function piece on its respective interval
  3. Use open or closed circles to indicate whether an endpoint is included (closed circle) or excluded (open circle) from an interval
• The domain of a piecewise function is the union of the domains of its individual pieces (combines all the intervals)
• The range of a piecewise function is the union of the ranges of its individual pieces determined by considering each function's behavior on its interval (combines all the y-values)

Notation for domain and range

• Interval notation represents a set of numbers using parentheses, brackets, and infinity symbols
  • Parentheses "(" and ")" indicate that the endpoint is not included in the interval (open interval)
  • Brackets "[" and "]" indicate that the endpoint is included in the interval (closed interval)
  • Examples: $(2, 5]$ represents all numbers between 2 and 5, including 5 but not 2; $(-\infty, 0)$ represents all numbers less than 0
• Set-builder notation defines a set by describing its elements using a variable and a condition
  • The general form is $\{x | \text{condition}(x)\}$, read as "the set of all x such that condition(x) is true"
  • Examples: $\{x | x \in \mathbb{R}, x > 0\}$ represents all real numbers greater than 0; $\{x | x \in \mathbb{Z}, -3 \leq x < 5\}$ represents all integers from -3 to 4, inclusive

Function Relationships and Properties

• Continuous functions have no breaks, gaps, or jumps in their graph
• The codomain is the set of all possible output values for a function, which may be larger than the actual range
• Mapping refers to the process of assigning each input value to a unique output value in a function
• Inverse functions "undo" the original function, swapping the roles of input and output