3.2 Domain and Range

Functions are all about relationships between inputs and outputs. In this topic, we dive into domains and ranges, which set the rules for what values can go in and come out of a function. Understanding these concepts helps us grasp how functions behave and what they can represent.

Real-world applications bring these ideas to life. We'll explore how domains and ranges apply to things like population growth, pricing, and motion. This practical angle shows how math concepts directly connect to everyday situations and problem-solving.

Domain and Range of Functions

Restrictions on input values

• Definition of domain: set of all possible input values (usually $x$) for which a function is defined
• Identifying restrictions on the domain
  • Denominator cannot equal zero in rational functions, find values of $x$ that make the denominator zero and exclude them from the domain
  • Square root of a negative number is undefined, ensure the radicand (expression under the square root) is non-negative for functions with square roots
  • Logarithms are only defined for positive arguments, the input must be greater than zero for logarithmic functions
• Notation for expressing domain
  • Interval notation: $[a, b]$ (closed interval), $(a, b)$ (open interval), $[a, b)$ (half-open interval), $(a, b]$ (half-open interval)
  • Set-builder notation: $\{x \mid \text{condition}\}$ defines the set of all elements $x$ that satisfy a given condition

Domain from function graphs

• Identifying domain from graphs: consists of all $x$-values for which the graph is defined
  • Vertical lines represent undefined values and are not part of the domain (asymptotes)
• Identifying range from graphs: consists of all $y$-values that the function takes on
  • Horizontal lines represent $y$-values that are not part of the range (asymptotes)
• Types of functions and their typical domains and ranges
  • Linear functions: domain and range are typically all real numbers ($\mathbb{R}$)
  • Quadratic functions: domain is all real numbers, range depends on the direction of the parabola (upward or downward)
  • Exponential functions: domain is all real numbers, range is typically positive real numbers ($\mathbb{R}^+$)
  • Logarithmic functions: domain is positive real numbers, range is all real numbers
• Continuous functions have a domain and range that form unbroken intervals

Real-world applications of domain

• Identifying the domain and range in context
  1. Determine what the input and output variables represent in the given context (time, distance, population)
  • Consider any real-world limitations on the input and output values (non-negative, integer values)
  • For discrete functions, the domain and range consist of specific, separate values
• Solving problems using domain and range
  • Use the domain to determine the allowable input values for the function in the given context
  • Use the range to interpret the possible output values and their meaning in the context
• Examples of real-world applications
  • Population growth models: domain is time (non-negative), range is population size (non-negative integers)
  • Supply and demand curves: domain is quantity (non-negative), range is price (non-negative)
  • Projectile motion: domain is time (non-negative), range is height (can be negative or positive)

Functions and Relations

• A function is a special type of relation where each input value corresponds to exactly one output value
• The codomain is the set of all possible output values for a function
• A relation is a set of ordered pairs that describes a relationship between two sets
• Mapping refers to the process of assigning output values to input values in a function