Chapter 7 Section 6 Functions

Learning Objective Find the Domain and range of a relation. Recognize functions Evaluate functions Graph linear functions

Key Vocabulary Relations Domain Range Function Vertical Line Test Function Notation Linear Functions

Find the Domain and Range of a Function Functions are a special type of relation and are common from algebra through calculus Relation is any set of ordered pairs and can have elements other than numbers. (x, y) ordered pairs x and y are the components of the ordered pair Every graph will represent a relation (candy, sales) (class, students) (birth year, life expectancy)(year, unemployment)

Find the Domain and Range of a Function (Domain, Range) Domain (x) of the relation is the first component of the ordered pair Range (y) of the relation is the second component of the ordered pair Think of a functions as a set of rules or directions that get you from one place (domain) to another (range)

Find the Domain and Range of a Function This example is a function because each person is assigned to exactly one seat. Sarah Brad Phillip Amanda Seat 1 Seat 2 Seat 3 Seat 4 DomainRange (Sarah, Seat 1) (Brad, Seat 2) (Phillip, Seat 3) (Amanda, Seat 4) Relation

Functions are ordered pairs where the first component (domain, x) corresponds with exactly one second component (range, y). First component is thought of as the input Second component is thought of as the output. For a function the output depends on the input. Garbage In = Garbage Out Find the Domain and Range of a Function

Do the following figures represent functions? Yes is a function Because the first components (domain) 1,2,& 3 only have one second component (range). As ordered pairs all the x’s are different ABAB (1, A) (2, A) (3, B) Domain (x) Range (y) Ordered Pair

Do the following figures represent functions? No, not a function Because the first component, domain (x), has two second components, range (y). 1 is associated with A and B As ordered pairs two have the same x. NOT a function 1212 A B C (1, A) (1, B) (2, C) Domain Ordered Pair Range

Find the Domain and Range of a Function Functions are useful because they allow us to predict the output when the input is known. Example Assume each candy bars cost $0.50 Write a function to determine the cost, c, when, n, candy bars are purchased. The function is c = 0.65n The cost (domain) depends on the number of candy bars purchased (range). One0.50(1)0.50 Two0.50(2)1.00 n0.50(n)0.50n

Determine which sets are functions? -6 2 Domain Range Example { (2,-3), (-6, 2), (2, 4)} No, not a function Because the first component, Domain (x) has two second components. 2 is assigned to -3 and 4 Domain: {-6, 2} Range: {-3, 2, 4}

Determine which sets are functions? Domain Range Example { (1, 3), (2, 4), (-2, 3)} 3434 Yes is a function Because the first components (1,2,& -2) only have one second component. Domain: {-2, 1, 2} Range: {3, 4}

Vertical Line Test If a vertical line can be drawn through any part of a graph and the vertical line intersects another part of the graph, then each value of x does not correspond to exactly one value of y and the graph does not represent a function. If a vertical line cannot be drawn to intersect the graph at more than one point, each value of x corresponds to exactly one value of y and the graph represents a function. x y Example: Points on the Graph. This is a FUNCTION because no vertical line Intersects more than one point

Vertical Line Test Example: Is not a function because one vertical line Intersects 3 points (1, 2) (1, -1) (1, -3) x y

Vertical Line Test Example: Function Not a Function

Vertical Line Test Example: Not a function Function

Evaluate Functions Graphs that we see every day represents function. Income, Sales Births, Year Function notation is f(x) reads as “f of x” Graph y = x + 2 is a function y depends on x Therefore, y = f(x) y is a function of the variable x y = f(x) = x + 2 f(x) = x + 2

Evaluate Functions Example: Let f(x) = x 2 – x + 2 Find f(-1) f(-1) = (-1) 2 – (-1) + 2 f(-1) = f(-1) = 4 When x = -1 f(x) or y = 4 NOTE: f(-1) is a shorthand for “Evaluate the expression that the function equals for x = -1”

Evaluate Functions Example: Let f(x) = x 2 – x + 2 Find f(3) f(3) = (3) 2 – f(3) = 9 – f(3) = 8 When x = 3 f(x) or y = 8 NOTE: f(3) is a shorthand for “Evaluate the expression that the function equals for x = 3”

Evaluate Functions Example: Let f(x) = x 2 – x + 2 If x = 4 find the value of y y = f(x) = x 2 – x + 2 y = f(4) = (4) 2 – y = f(4) = 16 – y = f(4) = 14 When x = 4 f(x) or y = 14 NOTE: f(4) is a shorthand for “Evaluate the expression that the function equals for x = 4”

Graph Linear Functions Remember that the graph of y = ax + b is a straight line that is a function y = mx + b and f(x) = mx + b are linear functions

Graph Linear Functions Example: Graph f(x) = -2x + 6 Let x = 3 (3, 0) f(3) = -2(3) + 6 f(3) = f(3) = 0 Let x = 2 (2, 2) f(2) = -2(2) + 6 f(2) = f(2) = 2 Let x = 1 (1, 4) f(1) = -2(1) + 6 f(1) = f(1) = 4 (3,0) (2,2) (1,4) Two way to graph 1.Using the Slope = -2 and y-intercept (0, 6) down 2 right 1 2. Plotting the points (0,6)

Graph Linear Functions Example: Ice Skating Rink, the weekly profits (p) on an ice skating rink is a function of the number of skaters per week (n). The function approximating the profit is p = f(n) = 8n – 600, where 0 ≤ n ≤ 400. What is the profit for 300 skaters? Graph f(n) = 8x Let n = 300 f(300) = 8(300) f(300) = f(300) = 1800 Let n = 200 f(200) = 8(200) f(200) = f(200) = 1000 Let n = 100 f(100) = 8(100) f(100) = f(100) =

Remember Think of a function as a set of rules or directions that get you from one place (domain) to another (range) Parentheses usually means multiply however, remember that function notation does not mean multiply Use the definition of a function to determine if an equation or a set of ordered pairs is a function. The vertical line test let you check if a graph is a function. What you do to one side of an equation, you do to the other side. When you put the value for x in the function notation on the left side of equal sign, you use the same value for x in the function definition on the right side

HOMEWORK 7.6 Page 476: # 35, 39, 45, 47, 57