Lesson 3.1

Warm-up Evaluate the following expressions: ÷ (3 – 1) – 2(2 + 4) 2 Challenge: 5. 6(5 – 7(-10 – -8) 2 )

Definitions Relations – how variables influence one another Functions – a relationship between variables in which each input has one and only one output. A function is always a relation, but a relation is not always a function. Domain – the set of inputs or x-values that can be evaluated for a particular function Range – the set of outputs or y-values that can be obtained for a particular function

What is a function? A function is defined as a relation where there is only one output for any given input. Example from our class: your birthday doesn’t change. No matter how often or at what time of day you ask your classmate for their birthday, the answer will stay the same. What are other examples of functions?

Examples of functions Functions can be found in a number of ways from coordinates to tables to graphs. What are you looking for to define a function? Every input has a distinct output. Graphically:Tabular: Coordinates: (2, 4), (5, 5), (-2, 6), (-7, 7) x F(x)4567

Functions vs. Relations Function - Ordered Pairs (1, 5), (2, 7), (3, 6), (2, 7) Each input has only one associated output Relation - Ordered Pairs (1, 5), (2, 7), (3, 6), (2, 9) At least one input has multiple outputs

Functions vs. Relations - Graph Passes Vertical Line Test - Graph Fails Vertical Line Test

Functions vs. Relations - Input/Output chart Each input has only one output. - Input/Output chart At least one input has more than one output

Functions vs. Relations Functions - x-y tables Each x has only one y Relations - x-y tables At least one x has more than one y xy xy

Function or not? The amount of money that the ticket taker at AMC makes in relation to the number of hours he works. The amount of money that a waitress at Chili’s makes in relation to the number of hours she works. The amount of money that a Chick-fil-A employee makes in relation to the number of hours worked.

Function or not? The heights of students in the class in relation to their student ID numbers. The heights of students in the class in relation to their shoe size. The heights of students in the class in relation to their birthday.

Example 1: {(3, 2), (4, 3), (5, 4), (6, 5)} function

Example 2: relation

Example 3: relation

Example 4: ( x, y) = (student’s name, shirt color) function

Example 5: Red Graph relation

Example 6 function Jacob Angela Nick Greg Tayla Trevor Honda Toyota Ford

Example 7 function A person’s cell phone number versus their name.

Function Notation y = x + 8 – You’ve seen this before! Remember these tables: Now, we simply replace y with f(x) where x is our input, and f(x) is the output that we’re trying to find. xy = x + 8y

Function Notation We can use any of the formulas we know for function notation. (Let’s try to stick with formulas that only have one variable) Example: the formula for the area of a circle is A = πr 2 In function notation, the formula becomes Area(r)= πr 2 or we can abbreviate it to A(r) = πr 2 What is our input in this function? What is our output? What are a couple other formulas we know that could be written in function notation?

Evaluating a function To evaluate a function in function notation, we simply replace the variable with the value. For example, the function f(x) = x + 5 when x = 4 We write that f(4) = (4) + 5 = 9 WE SIMPLY REPLACED THE “X” WITH ITS VALUE.

You Try Evaluate the Following Functions F(x) = 3x + 6G(x) = (4 – x) – 2 1. F(5) 2.F(-3) 3(5) + 6 = 213(-3) + 6 = G(5)4.G(-3) (4 – 5) – 2 = -3(4 - -3) – 2 = 5

Algebraic Functions F(x) = x + 5 F(x) = -3x + 2 F(x) = 1/x F(x) = √x Will any of the above expressions result in two answers for a given x?