Functions and Relations: An Introduction
Objectives: As I work through this investigation I will be able to:




Understand the difference between relations and functions
Know what a function looks like as a graph, data table or mapping diagram
Know and be able to use function notation
Understand what domain and range of a function looks like as a data table, mapping diagram and graph
New Concept: Relations
i. A "relation" is just a relationship between sets of information;
ii. A relation refers to a set of input and output values, usually represented in ordered pairs
iii. A relation is simply a set of ordered pairs.
Here are 4 everyday relations.
New Concept: y “ is a function of ” x
1. The height of water at the yacht club “ is a function of ” the time of the day.
2. The daily income of the canteen “ is a function of ” the number of customers.
3. The time of sunrise in Helsinki “ is a function of ” _________________________________
4. The number of bacteria in a pond “ is a function of ” ________________________________
Now you try..... Write 2 of your own.
5. ______________________________________________________________________________________
6. ______________________________________________________________________________________
Use two of your examples above and explain what you think y “ is a function of ” x means in your scenario means.
1.
2.
What would the graphs of these relations look like?
Scenario
Independent Variable:
The height of water at the yacht club
“ is a function of ” the time of the
day.
Dependent Variable:
Is this a function? Explain
Sketch the Graph
Scenario
Independent Variable:
Sketch the Graph
The daily income of the canteen “ is a
function of ” the number of
customers.
Dependent Variable:
Is this a function? Explain
Scenario
Independent Variable:
Sketch the Graph
The time of sunrise in Helsinki “ is a
function of ”
Dependent Variable:
Is this a function? Explain
Scenario
Independent Variable:
Sketch the Graph
The number of bacteria in a pond “ is
a function of ”
Dependent Variable:
Is this a function? Explain
Check in: Check you understanding with your midnight partner. Discuss any inconsistencies you may have and agree as
a pair.
New Concept: Functions
i.
A function is a "well-behaved" relation  When we say that a function is "a well-behaved relation", we mean that,
given a starting point, we know exactly where to go; given an x, we get only and exactly one y.
ii.
A function is a set of ordered pairs in which each x-element has only ONE y-element associated with it.
iii.
A function is a rule that takes an input, does something to it, and gives a unique corresponding output.
Exploration: What do Functions Look like?
Scenario 1
Scenario 2:
Reflection: Look at the 2 scenarios above to answer these questions. Consider the definitions of functions.
1. How can I tell a relation is a function from a table-of-values? Draw an example of a function and a non-function.
2. How can I tell a relation is a function from a mapping diagram? Draw an example of a function and a non-function.
3. How can I tell a relation is a function from a graph? Sketch an example of a function and a non-function.
Function
Not a Function (a Relation only)
Check For Understanding:
For each of the following, determine if it is a function or a relation, justify your answer.
Text Reference:
Page 270 Exercise 9A - Data tables and Graphs
New Concept: Notation of Functions
I.
Rather than writing linear equations in the typical y = mx + b format, we will now write them in function notation 
as f(x) = mx + b where “f ” simply refers to the function name (rule) and the ‘x’ refers to the input.
II. Evaluate  if f(x) = 2x + 4, then we can evaluate f(3) as ……
III.
Solve  if f(x) = 2x + 4, then we can solve 12 = f(x) as …..
Example:
From the mapping diagram determine:
If f(x) = 24, find x.
Explain how you did this.
Example:
From the graph determine:
f(0) =
f(3) =
If f(x) = 1, find x.
If f(x) = -7, find x.
Example:
From the data table:
x
0
1
2
3
4
5
f(0) =
f(x)
2
5
8
2
11
-1
f(3) =
If f(x) = 11, find x.
If f(x) = 2, find x.
Check for Understanding: Do these questions
Text: page 275 Exercise 9C – Q. 1, 3, 4, 11, 12
What does this tell us about f(x)?
New Concept: Domain and Range
i.
The set of all the starting points is called "the domain" and the set of all the ending points is called "the range."
ii.
The domain is what you start with; the range is what you end up with.
iii.
The domain is the x's; the range is the y's
Example: Mapping Diagram
From the mapping diagram example above determine:
Domain: {2, 5, 10}
Range: ____________________________
Example: Data Table
x
0
1
2
3
4
5
f(x)
2
5
8
2
11
-1
From the data table determine:
Domain: __________________________
Range: ____________________________
Example: Graph
From the graph determine:
Domain: __________________________
Range: ____________________________
How is this question different from other examples?
Check for Understanding – Determine the Domain and Range of these relations.
Text Practice – Page 273 Exercise 9B Q. 1