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1.1 Relations and Functions
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MCR3U Unit 2 – Intro to Functions Date: 1.1 Relations and Functions
Homework: Study for the Unit 1 Equivalent Algebraic Expressions (2.1-‐2.7) Test! Learning Objectives/Success criteria: At the end of this lesion I will be able to:
o Use set notation o Differentiate between a relation and a function o Identify the domain and range o Identify a function from:
o Words o Tables of values
o Sets of ordered pairs o Mapping diagrams
o Graphs o Equations
o Use the vertical line test
Set Notation: A set is a collection of things or elements listed inside braces or curly brackets { }.
and we could name that set using capital letters. We can also use a special character ∊ to say something is an element of a set.
A relation is a set of ordered pairs. Values of the independent variable are paired with values of the dependent variable. Ordered pairs are written in parentheses or round brackets ( ) with the independent (input) variable first and the dependent (output) variable(s) following. The parts of an ordered pair can be broken up into sets of its independent and dependent variable. The domain is a set of the independent variable of a relation. For example: all the possible “x” values. The range is the set of the dependent variable of a relation. For example: all the possible “y” values. Both sets of the domain and range are ordered meaning they are written in ascending order. Example 1: Consider the set A={(0,1), (-‐1,2), (2,2), (-‐3, 5)}: Domain = Range = Example 2: Write a set of ordered pairs describing which month 5 students were born in: What is the domain and range of this relation?
A function is a relation where one value of the independent variable corresponds with only one value of the dependent variable. Identifying functions:
1. Words: a) if x=1, y=3 or 4 b) if x=0, y=6 and 2 c) if x=3, y=8 d) if x=1, y=9 if x=5, y=1 if x=4, y=8
2. Table of Values: x y -‐2 4 -‐1 1 0 0 1 1 2 4
3. Sets of Ordered Pairs:
A = { −3,0( ), (−1,1), (0,1), (4, 5), (0, 6)} B = {(−2,1), (1,1), (0, 0), (2,−2)}
4. Mapping Diagrams: The elements of the domain are mapped onto the elements in the range using
arrows. If each element in the domain has only one corresponding element in the range, the relationship is a function.
x y -‐2 4 -‐1 1 -‐1 0 1 1 -‐2 6
5. Graphs: We can use the vertical line test to determine graphically if a relation is a function. If any
vertical line intersects the graph of a relation more than once, then the relation is not a function. If the vertical line intersects the graph at more than one point, it shows that there are x-‐values in the domain that correspond to two y-‐values in the range. a) Discrete Data b) y = x2
{(1,3) , (2,-‐2) , (1,4) , (6,3) , (0,5)}
c) y = 2x +1 d) x2 + y2 = 9
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e) f) − x + 2 g) h) sin(x)
6. Equations: We can substitute values for our independent variable and solve for our dependent variable. If we get more than one answer for our dependent variable, then the relationship is not a function. a) y = 2x − 5 b) x2 + y2 = 9
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