Basics of Functions and Their Graphs Objectives Identify the domain and range of relations and functions. Determine whether a relation is a function.

Vocabulary relation domain range function

A relation is a pairing of input values with output values A relation is a pairing of input values with output values. It can be shown as a set of ordered pairs (x,y), where x is an input and y is an output. The set of input values for a relation is called the domain, and the set of output values is called the range.

(x, y)  (input, output)  (domain, range) Mapping Diagram Domain Range A 2 B C Set of Ordered Pairs {(2, A), (2, B), (2, C)} (x, y)  (input, output)  (domain, range)

Identifying Domain and Range Give the domain and range for this relation: {(100,5), (120,5), (140,6), (160,6), (180,12)}. List the set of ordered pairs: {(100, 5), (120, 5), (140, 6), (160, 6), (180, 12)} Domain: {100, 120, 140, 160, 180} The set of x-coordinates Range: {5, 6, 12} The set of y-coordinates

Give the domain and range for the relation shown in the graph. List the set of ordered pairs: {(–2, 2), (–1, 1), (0, 0), (1, –1), (2, –2), (3, –3)} Domain: {–2, –1, 0, 1, 2, 3} The set of x-coordinates. Range: {–3, –2, –1, 0, 1, 2} The set of y-coordinates.

Suppose you are told that a person entered a word into a text message using the numbers 6, 2, 8, and 4 on a cell phone. It would be difficult to determine the word without seeing it because each number can be used to enter three different letters.

Number {Number, Letter} {(6, M), (6, N), (6, O)} The numbers 6, 2, 8, and 4 each appear as the first coordinate of three different ordered pairs. {(2, A), (2, B), (2, C)} {(8, T), (8, U), (8, V)} {(4, G), (4, H), (4, I)}

However, if you are told to enter the word MATH into a text message, you can easily determine that you use the numbers 6, 2, 8, and 4, because each letter appears on only one numbered key. {(M, 6), (A, 2), (T, 8), (H,4)} In each ordered pair, the first coordinate is different. The “x-coordinate” never repeats. A relation in which the first coordinate is never repeated is called a function. In a function, there is only one output for each input, so each element of the domain is mapped to exactly one element in the range.

A relation in which each member of the domain corresponds to exactly one member of the range is a function. Notice that more than one element in the domain can correspond to the same element in the range. Aerobics and tennis both burn 505 calories per hour. Is this relation a function? No

Determine whether each relation is a function? {(1, 8), (2, 9), (3, 10)} Yes {(3, 4), (5, 6), (3, 7)} No {(2, 6), (3, 6), (4, 6)} Yes

Functions as Equations

Here is an equation that models paid vacation days each year as a function of years working for the company. 𝒚=−𝟎.𝟎𝟏𝟔𝒙 + .𝟗𝟑𝒙+𝟖.𝟓 The variable x represents years working for a company. The variable y represents the average number of vacation days each year. The variable y is a function of the variable x. For each value of x, there is one and only one value of y. The variable x is called the independent variable because it can be assigned any value from the domain. Thus, x can be assigned any positive integer representing the number of years working for a company. The variable y is called the dependent variable because its value depends on x. Paid vacation days depend on years working for a company.

Not every set of ordered pairs defines a function. Not all equations with the variables x and y define a function. If an equation is solved for y and more than one value of y can be obtained for a given x, then the equation does not define y as a function of x. So the equation is not a function. Determine whether each equation defines y as a function of x. Hint: Solve for y. 𝑥+4𝑦=8 𝑥 2 +2𝑦=10 𝑥 2 + 𝑦 2 =16 Yes Yes No

Function Notation

The special notation 𝑓(𝑥), read as "𝑓 𝑜𝑓 𝑥" represents the value of the function at the number, 𝑥. If a function named 𝑓, and 𝑥 represents the independent variable, the notation 𝑓(𝑥) corresponds to the y-value for a given 𝑥. The special notation 𝑓(𝑥) is a fancy way of expressing the dependent variable, 𝑦. 𝒇 𝒙 =−𝟎.𝟎𝟏𝟔 𝒙 𝟐 + .𝟗𝟑𝒙+𝟖.𝟓 or 𝒚=−𝟎.𝟏𝟔 𝒙 𝟐 + .𝟗𝟑𝒙+𝟖.𝟓 Evaluate 𝑓 10 . (Hint: substitute 10 for every x variable.)

Graphing Calculator- evaluating a function Press the Y = key. Type in the equation −𝟎.𝟏𝟔 𝒙 𝟐 + .𝟗𝟑𝒙+𝟖.𝟓 Quit this screen by pressing 2nd Mode (Quit). Press the VARS key. Move the cursor to the right to Y-VARS. Press ENTER on 1 (Function) Press ENTER on Y. Type (10), then ENTER. .

Evaluate each of the following. Find f(2) for 𝑓 𝑥 =2 𝑥 2 +4 Find f(−2) for 𝑓 𝑥 =9 − 𝑥 2 Find f(x + 2) for 𝑓 𝑥 = 𝑥 2 −2𝑥+4 Find f(−x) for 𝑓 𝑥 = 𝑥 2 −2𝑥+4

The Vertical Line Test of a Function If any vertical line intersects a graph in more than one point, the graph does not define y as a function of x.

Use the vertical line test to identify graphs in which y is a function of x.

Obtaining Information From Graphs

Identifying Domain and Range from a Function’s Graph Using Interval Notation Find the coordinates of the endpoints.

(𝟑, 𝟎) Find the coordinates of the endpoints. Domain: (−∞, 3] Range: [0, ∞)

(−𝟒, 𝟐) (𝟑, 𝟎) Find the coordinates of the endpoints. Domain: [−4, 3) Range: (0, 2]

Domain: [−2, 4) Range: [1, 3]

Identifying Intercepts from a Function’s Graph

𝑥 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 𝑖𝑠 −3 𝑓 −4 =2

𝑦 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡=2 𝑓 2 =3

𝑥 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡𝑠=0 𝑎𝑛𝑑 4 𝑓 5 =4 𝑦 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡=0