Download 2.1-2.6 Notes: Relations and Functions

Notes: Relations and Functions
Learning Targets:
2.1- I can describe ordered pairs using the coordinate system, a table, and mapping.
2.2- I can analyze the domain and range of relations and functions.
2.3- I can determine if a relation is a function.
2.4- I can explain the parts of a Cartesian plane in respect to relations and functions. (Throughout lesson)
2.5- I can create multiple representations of relations and functions.
2.6- I can evaluate functions.
 Coordinate Plane: grid formed by the intersection of two number lines, the horizontal axis (x-axis) and the
vertical axis (y-axis).
 Ordered Pair: set of numbers, or coordinates, written in the form (x, y).
 X-Coordinate/X-Value/Input/Domain: first number in each ordered pair.
 Y-Coordinate/Y-Value/Output/Range: second number in each ordered pair.
 Independent Variable: (Domain/x-coordinates) the variable that determines the output.
 Dependent Variable: (Range/y-coordinates) the variable with a value that depend on the value of the
independent variable. (Remember: y depends on x)
 Coordinate Plane/Cartesian Plane: Label each quadrant, the x and y axis, and the origin.

Write the ordered pair for each point shown in the coordinate plane below (point a thru point k). Then name
the quadrant in which each point is located.
 Relation: a set of ordered pairs.
 A Relation can be represented in several ways:
o
o
o
o
 Representations of Relations
Ordered Pair:
Table:
(-1, 4)
(0, 2)
(2, 5)
(5, 1)
Graph:
Mapping: Illustrates how each element of the
Domain is paired with an element in the range.
 Practice: Create a table, graph, and mapping for the following relation below.
o { (2, -5), (3, 2), (-1, 4), (0, -1) }
Table:
Graph:
Mapping:
 Domain: ________________________________________________________________________
 Range: _________________________________________________________________________
Example: {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3)}
Domain: {1, 2}
Range: {1, 2, 3}
 State the domain and range of the relations below.
1. {(2, 1), (4, 2), (3, 3), (4, 1)}
2. {(1, 4), (2, -3), (-2, 1), (0, 3)}
3. {(2, 0), (-1, 3), (2, 1), (1, 3)}
D=
D=
D=
R=
R=
R=
 Function: A relation in which each element of the domain is paired with exactly one element of the range.
 Focus on the x-coordinates, when given a relation
 If the set of ordered pairs have different x-coordinates, it IS A function.
 If the set of ordered pairs have same x-coordinates, it is NOT a function.
 Y-coordinates have no bearing in determining functions
 ***If the relation is a function, then the domain doesn’t share.
 A function can be represented as a:
o
o
o
o
o
 Representations of Functions
-Create a table, set of ordered pairs, mapping, and a graph for the equation below.
Equation:
𝑦 = 2𝑥 + 2
*To graph any function (equation)1.
2.
3.
Make a table of values (X/Y table)
made up of chosen X-values.
Substitute your X-values into the
equation to find the
corresponding Y-values.
Graph
Table:
Ordered Pairs:
Mapping:
Graph:
 Identifying Functions: Determine whether the following is a function.
Ex.1) {(0, -5), (1, -4), (2, -3), (3, -2), (4, -1), (5, 0)}
Ex.2) {(-1, -7), (1, 0), (2, -3), (0, -8), (0, 5), (-2, -1)}
Ex.3)
Ex.4)
Ex.5)
Ex.6)
X
-5
-2
4
2
X
-2
3
1
3
Y
2
-1
2
0
Y
-5
3
4
1
 Vertical Line Test- a relation is a function if a vertical line intersects the graph only once.
 If the vertical line intersects the graph more than once, the graph is NOT a function.
 AKA: “The Pencil Test”- Take a pencil and move it from left to right (–x to x); if it crosses more than one
point, it is not a function
 Use the vertical line test to determine if the following graphs represent a function.
Example 1:
Example 2:
Example 3:
 Real World
 Example: Which situation represents a function?
a) The items in a store to their prices on a certain date.
b) Types of fruits to their colors

Extra Practice: Determine if relations below are functions?
1. Names and social security numbers.
2. Addresses and names.
 Discrete Function- A graph that consists of points that are not connected.
 Continuous Function- A function graphed with a line of smooth curve.
 Determine if the following functions are discrete or continuous. Then identify the domain and range.
Ex. 1)
Ex. 2)
Type of Graph: ______________
Type of Graph: _______________
Domain: ___________________
Domain: ____________________
Range: ____________________
Range: _____________________
Ex. 3)
Ex. 4)
Type of Graph: ______________
Type of Graph: _______________
Domain: ___________________
Domain: ____________________
Range: ____________________
Range: _____________________
 EVALUATING FUNCTIONS:
 Function Notation- Equations that are functions can be written in function notation.
Equation
Function Notation
Y = 3x - 8
f(x) = 3x - 8
 𝒇(𝒙) means function of 𝑥 and is read “𝒇of 𝒙.”
 In a function, 𝒙 represents the elements of the domain, and 𝒇(𝒙) represents the elements of the range.
 Ex.) Suppose you want to find the value in the range that corresponds to the element 1 in the domain.
EX) For 𝑓(𝑥) = 2𝑥 + 1, find the value of 𝑓(1).
The notation f(1) means to replace x with 1.
𝑓(𝑥) = 2𝑥 + 1
𝑓(1) =
 Ex.) Given 𝑔(𝑥) = 𝑥 2 − 3, find 𝑔(−2).
 Given 𝑓(𝑥) = 2𝑥 2 − 3𝑥, find the following.
a) 𝑓(3)
 Extra Practice: Evaluate the following.
1. For 𝑓(𝑥) = −4𝑥 + 7, find 𝑓(2).
b) 3(𝑓(𝑥))
2. For 𝑔(𝑥) = 2𝑥 2 − 5, find 𝑔(−3).
3. For 𝑓(𝑥) = 3𝑥 + 2, find 𝑓(4) − 5.
4. For 𝑔(𝑥) = −2𝑥 + 4, find 𝑔(ℎ + 5)
 Real World Connection:
The table below shows the masses, m grams, of different numbers of identical marbles, n.
Number of Marbles (n) Mass of Marbles (m grams)
1
1.27
2
2.54
3
3.81
4
5.08
5
6.35
6
7.62
A.
B.
C.
D.
Why is the relation also a function?
Identify the independent variable and the dependent variable. Justify your choices.
Create a set of ordered pairs, a mapping, a graph, and an equation for the function above.
Using the equation you created in part C, find the mass of 12 marbles, or 𝑓(12).
A. _________________________________________________________________________________________
___________________________________________________________________________________________
B. __________________________________________________________________________________________
____________________________________________________________________________________________
C. Ordered Pairs
D. 𝑓(𝑥) =
𝑓(12) =
Mapping
Graph
Equation