# Functions and Graphs

## Overview

### Description

A function is a relationship between two sets of numbers: the domain and range. Each number in the domain maps to a unique value in the range. Functions may be represented in different forms, including tables, graphs, and equations. The graph of a function can be used to find information about the function, such as the domain, range, maximum values, and minimum values. Graphs of functions may be transformed by translations, stretches, compressions, and reflections.

### At A Glance

• A function is a relation between two sets called the domain and range, in which each element of the domain corresponds to exactly one element of the range. The relationship can be represented by a mapping diagram, an algebraic rule, or a graph.
• The domain and range of a function are the sets of values that define a function. The domain is all possible inputs to the function. The range is all possible outputs from the function.
• A function can be evaluated for a specific element in the domain by finding the corresponding element of the range.
• Functions can be added, subtracted, multiplied, and divided.
• A graph of a function is a visual representation of the function. An algebraic rule can be used to produce the graph of a function.
• The vertical line test uses vertical lines to determine whether a relation is a function.
• A piecewise function consists of separate pieces of the same function. Each piece behaves differently based on the rules of their defined intervals.
• Functions can be identified as even, odd, or neither. An even function is a line of symmetry about the $y$-axis. An odd function has rotational symmetry about the origin.
• Determining whether a graph is increasing, decreasing, or constant depends on how the $x$- and $y$-values increase, decrease, or remain the same.
• A function's maxima and minima are determined by the lowest and highest points at specific intervals, called local minimum and local maximum, as well as the function's highest point, called the global maximum, and its lowest point, called the global minimum.
• The most basic function from a family of functions is called a parent function. Related functions can be graphed by modifying the graph of the parent function.
• The graph of a function can be translated vertically or horizontally by performing addition or subtraction within the function rule.
• The graph of a function can be stretched or compressed vertically or horizontally by performing multiplication within the function rule by a positive constant.
• The graph of a function can be reflected across the $x$- or $y$-axis by performing multiplication within the function rule by –1.
• The graph of a function can be transformed by using a combination of translations, stretches, compressions, and reflections.