Relations and Functions: Domain, Range, and Graphical Analysis

Module 1 Topic 2: Relations & Functions

Learning Objectives Overview

This module introduces foundational concepts in College Algebra related to relations and functions. Students will learn to analyze, evaluate, and graph functions, as well as interpret their properties from equations and graphs.

Relations and Functions

A relation is a set of ordered pairs, while a function is a special type of relation where each input has exactly one output.

• Domain: The set of all possible input values (usually x-values).

• Range: The set of all possible output values (usually y-values).

• Function: A relation in which each element of the domain is paired with exactly one element of the range.

Example: The relation {(1,2), (2,3), (3,4)} is a function because each input has one output.

Determining Functions from Equations

To determine if an equation represents a function, check if each input value yields only one output value.

• Vertical Line Test: A graph represents a function if no vertical line intersects the graph at more than one point.

Example: The graph of $y = x^2$ passes the vertical line test and is a function.

Evaluating Functions

To evaluate a function, substitute a given input value into the function's equation.

• Notation: If $f(x) = 2x + 3$, then $f(4) = 2(4) + 3 = 11$.

Graphing Functions by Plotting Points

Functions can be graphed by calculating output values for selected inputs and plotting the resulting points.

• Choose several x-values from the domain.

• Calculate corresponding y-values using the function.

• Plot the points $(x, y)$ on a coordinate plane.

Example: For $f(x) = x^2$, plot points for $x = -2, -1, 0, 1, 2$.

Using the Vertical Line Test

The vertical line test is a graphical method to determine if a curve is the graph of a function.

• If any vertical line crosses the graph more than once, the graph does not represent a function.

Obtaining Information from Graphs

Graphs can provide information about a function's behavior, such as intercepts, increasing/decreasing intervals, and maximum/minimum values.

• x-intercept: Where the graph crosses the x-axis ($y=0$).

• y-intercept: Where the graph crosses the y-axis ($x=0$).

Identifying Domain and Range from Graphs

The domain and range of a function can often be determined by examining its graph.

• Domain: The set of all x-values for which the graph exists.

• Range: The set of all y-values that the graph attains.

Example: For the graph of $y = x^2$, the domain is all real numbers, and the range is $y ext{ such that } y \\geq 0$.