Algebra

Linear functions

Linear functions are an important topic in the study of algebra. They are also used to describe many real-world situations, such as the relationship between distance, speed, and time, or the relationship between revenue, price, and quantity. In fact, there are so many applications that rely on linear functions that the topic is often embedded in many other subject areas such as science, business, geography, computer science, and so on. Before diving into linear functions it is beneficial to take some time to compare and contrast the properties and definitions for mathematical relations and functions.

Objectives

Identify relations and functions.
Find domain and range.
Use the five ways for representing functions.

Relations

A relation is a pairing of two elements. Relations can be expressed using ordered pairs, such as (x, y). In mathematics we expect ordered pairs to be the pairing of numeric values, such as (1, 2). However, they can be the pairing of any two pieces of data. When you have a collection of ordered pairs they form a relation.

The first element of an ordered pair is called the input and comes from the domain of the relation. The second element of an ordered pair is called the output and comes from the range of the relation.

{ (input , output) }

{ (domain , range) }

Ex1: A relation of U.S. states to their senators would look like:

{ (NJ , Cory Booker) ; (NJ , Bob Menendez) ; (PA , Pat Toomey) ; (NY , Chuck Schumer) }

Domain = {NJ , PA , NY}

Range = {Cory Booker , Bob Menendez , Pat Toomey , Chuck Schumer}

Ex 2: A relation of automotive makes and models:

{ (Ford, Mustang) ; (Chevy, Camaro) ; (Chevy , Corvette) ; (Dodge, Challenger) }

Domain = {Ford, Chevy, Dodge}

Range = {Mustang, Camaro, Corvette, Challenger}

Relations permit the pairing of multiple elements from the range with the same element from the domain. However, functions do not.

Functions

A function is a relation with the property that each element of the domain is paired to exactly one element in the range.

Examples of Functions:

Ex 3: A collection that pairs Student ID’s with Student Names.

Ex 4: A collection that pairs Planets with the number of moons.

In both Ex 3 and Ex 4, the input values are unique and never repeat. The output values on the other hand may or may not repeat and that is okay for the description is still representative of a function.

Five Ways to Represent Relations and Functions

Set Notation
Mappings
Tables
Graphs
Equations

Set Notation

Sets are collections of data usually contained as lists within braces { }. When relations and functions are represented as a lists, they often are depicted as ordered pairs within braces, such as:

Mappings

A mapping is an arrow diagram that links the elements of the domain to the elements of the range. The image shown can be used to create a set of ordered pairs and determine whether the mapping is a relation or a function.

The set of order pairs that come from this mapping are:

{ (2, 11) ; (1, 7) ; (1, 14) ; (3, 9) }

Tables

Tables are useful for organizing data into ordered pairs. When discussing relations and functions it is common to name the first column x and the second column y. Recall that x represents the input values which come from the domain and y represents the output values which come from the range.

Graphs

A graph is a picture representation of a collection of ordered pairs . We plot the points on the coordinate plane with a horizontal axis, called the x – axis, which represents the domain, and a vertical axis, called the y – axis, represents the range. Graphical representations provide a visual that helps us better understand the difference between a relation and a function. Recall, that relations permit the pairing of multiple y – values to each x – value but functions do not. This can be illustrated using a vertical line.

The graph below contains a set of ordered pairs that can be classified as a relation. It is a relation because at least one input value has multiple corresponding output values. Consider the input value x = 3, notice that it has more than one corresponding output value for y, as shown by the points (3, 5) and (3, 10). This concept can be demonstrated graphically using the Vertical Line Test. If a vertical line can touch more than one point anywhere in the graph, then it is a relation. If this were a function, then there would be no place in the graph that a vertical line could intersect two points or more.

Equations

Equations are used to generate output values that correspond to the input values from the domain. Typically, equations are written using “y – equals” form, but since we are studying equations in the context of functions the y – variable is replaced by function notation. For example, y = 2x – 1 is expressed as a function by writing f (x) = 2x – 1, where f (x) replaces y.

Comments:

· f (x) is the function name and is read as “f of x”

· Function names are useful when handling multiple equations. A population function could be named P (x), a revenue functions could be named R (x), and so on.

· For x = c , where c is a real number, f (c) is the resulting y – value when c is plugged into the equation for x.

· An ordered pair can be written (x, f (x)) or as (x, y), they are the same.

Ex 5: Given the function equation f (x) = 3x – 2, evaluate f (5) .

Evaluate f (5) means evaluate the expression 3x – 2 for x = 5 .

The expression becomes 3(5) – 2 which evaluates to 13.

Therefore f (5) = 13 .

Ex 6: Create a table of values for the function g(x) = 4x + 1using the following x – values.

x = { – 2 , –1 , 0 , 1 , 2}

Create a table with two columns, but instead of labeling the columns x and y, name them x and g(x). Populate the column for x with the given values.

x = {– 2 , –1 , 0 , 1 , 2}

Calculate each corresponding y – value as shown in red and put the result in the g(x) column.

Page updated

Google Sites

Report abuse