Functional Relationships

Functional Relationships

Functional Relationships Day 1

Vocabulary: • A function is a relation in which each element of the domain is paired with ___________element of the range. Another way of saying it is that there is ______________output (y) per input (x). f(x) y x

Sketch a linear function. Sketch a nonlinear function. • Non-Linear Function: • Does not make a line. • Linear Function: • Makes a line

How about some more definitions? Thedomain is the x or input value in a function. (set of 1st coordinates of the ordered pairs) (2, 0) or y = 3x + 2 The range is the y or output value in a function. (set of 2nd coordinates of the ordered pairs) (2, 0) ory = 3x + 2 A relation is a set of ordered pairs.

Given the relation {(3,2), (1,6), (-2,0)}, find the domain and range. Domain = Range =

The relation {(2,1), (-1,3), (0,4)} can be shown by 1) a table. 2) a mapping. 3) a graph. x y 2 -1 0 1 3 4 2 -1 0 1 3 4

How can you tell if a relation is a function without a graph? • Only ONE output per input • Coordinates: Check all x values. ___ can not be repeated • Mapping: Can only have ________ drawn from each x • Graph: passes _______________ test

Given the following table, show the relation, domain, range, and mapping.x -1 0 4 7y 3 6 -1 3 Relation = {(-1,3), (0,6), (4,-1), (7,3)} Domain = Range =

Mappingx -1 0 4 7y 3 6 -1 3 You do not need to write 3 twice in the range! 3 6 -1

What is the domain of the relation{(2,1), (4,2), (3,3), (4,1)} • {2, 3, 4, 4} • {1, 2, 3, 1} • {2, 3, 4} • {1, 2, 3} • {1, 2, 3, 4}

What is the range of the relation{(2,1), (4,2), (3,3), (4,1)} • {2, 3, 4, 4} • {1, 2, 3, 1} • {2, 3, 4} • {1, 2, 3} • {1, 2, 3, 4}

Vertical Line Test (pencil test) If any vertical line passes through more than one point of the graph, then that relation is not a function. Are these functions?

Vertical Line Test FUNCTION! NO! NO WAY! FUNCTION!

Other Related Vocabulary: • Independent Variable (input): the variable that determines the value of the dependent variable. (x axis or domain values) • Dependent Variable (output): The variable relying on the independent variable (y axis or range values) EXAMPLE: the diameter of a pizza and its cost

Finding Domain and Range of a Graph • First identify all possible values for the domain (x or input). • Next, identify all possible values for the range (y or output). x values: -9 through +8 which can be written as: -9 ≤ x ≤ 8 RANGE DOMAIN y values: -3 through +8 which can be written as: -3 ≤ y ≤ 8

Practice: Finding the Domain and Range of a Graph • First identify all possible values for the domain (x or input). • Next, identify all possible values for the range (y or output). x values: -5 through +6 which can be written as: -5 ≤ x ≤ 6 RANGE DOMAIN y values: -4 through +7 which can be written as: -4 ≤ y ≤ 7 IS THIS A FUNCTION??

Relations & Functions-YEAR 1 A function is like a machine. You put something in and you get something out. Input x Sometimes equations have two variables. When there are two variables in the equation, all solutions are ordered pairs. (x, f(x)) There are an infinite number of solutions for a two variable equation. Rule f(x) Output

Function Notation • For example, with a function f(x) = 2x, if the input is 5, then it is written as f(5) = 2(5) • The output is ____. Input 5 2x 10 2(5) 10 Output

INPUT Human Years RULE OUTPUT Dog years x 7x f(x) EXAMPLE: Complete the table to find out the human ages of dogs ages 3 through 6. 7(3) 3 21 4 7(4) 28 5 7(5) 35 6 7(6) 42 So, a 3 year old dog is 21 in human years … 4 year old dog is 28 … … 5 year old dog is 35 … … 6 year old dog is 42 …

EXAMPLE: Make a function table to find the range of f(x) = 3x + 5 if the domain is {-2, -1, 0, 3, 5}. x 3x + 5 f(x) -2 3(-2) + 5 -1 -1 3(-1) + 5 2 0 3(0) + 5 5 3 3(3) + 5 14 5 3(5) + 5 20 Range: {-1, 2, 5, 14, 20}.

More Examples EXAMPLE: Find f(-3) if f(n) = -2n – 4 f(-3)= -2(-3) – 4 f(-3) = 2 EXAMPLE: Find

Functional Relationships