Functions
Lesson 2
Warm Up

1. Write an equation of the line that passes
through the points (-2, 1) and (3, 2).
x  5y  7  0

2. Find the gradient of the line that is
perpendicular to the line 4x – 7y = 12.
7
m
4

3. Write the equation of the vertical line that
passes through the point (3, 2).
x3
Relation

Relation – pairs of quantities that are related
to each other

Example: The area A of a circle is related to
its radius r by the formula
A  r .
2
Function

There are different kinds of relations.

When a relation matches each item from one
set with exactly one item from a different set
the relation is called a function.
Definition of a Function

A function is a relationship between two
variables such that each value of the first
variable is paired with exactly one value of
the second variable.

The domain is the set of permitted x values.

The range is the set of found values of y.
These will be called images.
Let’s take a look at the function that relates
the time of day to the temperature.
Rules to be a Function
Is it a Function?


For each x, there is
only one value of y.
Therefore, it IS a
function.
Domain, x
1
2
3
4
5
6
52
Range, y
-3.6
-3.6
4.2
4.2
10.7
12.1
52
Is it a function?


Three different yvalues (7, 8, and 10)
are paired with one xvalue.
Therefore, it is NOT a
function
Domain, x
3
3
3
4
10
11
52
Range, y
7
8
10
42
34
18
52
Function?

Is it a function? Name the domain and range.
{(3, -5), (4, -8), (5, 6), (7, 10), (8, 2)}



YES. For every x-value, there is only one
value of y.
Domain: (3, 4, 5, 7, 8)
Range: (-5, -8, 6, 10, 2)
Function?

Is it a function? State the domain and range.
{(5, 8), (6, 7), (3, -1), (4, 2), (5, 9), (12, -2)



No. The x-value of 5 is paired with two
different y-values.
Domain: (5, 6, 3, 4, 12)
Range: (8, 7, -1, 2, 9, -2)
Function?

Is it a function? Name the domain and range.
{(-2, 3), (4, 6), (3, 1), (7, 6), (9, -3), (2, 8)}



Yes. For every x-value, there is only one
value of y.
Domain: (-2, 4, 3, 7, 9, 2)
Range: (3, 6, 1, -3, 8)
Function?
YES
Vertical Line Test

Used to determine if a graph is a function.

If a vertical line intersects the graph at more
than one point, then the graph is NOT a
function.
NOT a function
IS a function
You Try…...
You Try….
You Try: Is it a Function?

YES
You Try…Is it a function?

YES.
You Try…Is it a Function?

NO.
Is it a function? Give the domain and range.
FUNCTION
Domain :  4,2
Range :  4,4
Give the Domain and Range.
Domain : x  1
Range : y  2
Domain : 2  x  2
Range : 0  y  3
IB Notation….

When a function is defined for all real values,
we write the domain of f as
Functional Notation

We have seen an equation written in the form
y = some expression in x.

Another way of writing this is to use
functional notation.

For Example, you could write y = x²
as f(x) = x².
Functional Notation
f(x) = 3x + 5
Find:
f (2)
3 2  5
65
1
f (0)
f (5)
30   5
35  5
05
15  5
5
20
Functional Notation
f ( x)  3 x  x  2
2
Find:
f (3)
3 3   3  2
27  3  2
30  2
32
2
f (0)
f (4)
30   0  2
002
34   4  2
316   4  2
48  4  2
46
2
2
2
Functional Notation
f ( x)  x  x  2
2
Find:
f (m)
m m2
2
f ( m  3)
m  3  m  3  2
m  3m  3  m  3  2
2
m 2  3m  3m  9  m  3  2
m  5m  8
2
Let’s look at Functions
Graphically
f ( 2)  g ( 4)
Find:
f ( x)
g ( x)
Find:
f ( 1)  g ( 0)
f ( x)
g ( x)
Find:
f ( 2)  g ( 1)
f ( x)
g ( x)
f (5)  g (0)
Find:
f ( x)
g ( x)
Find:
f ( 4)  g ( 1)
f ( x)
g ( x)
Find:
f ( 4 )  g ( 2 )
f ( x)
g ( x)
Find:
f ( 2)  g (0)
f ( x)
g ( x)
Find:
g (5)  f ( 3)
f ( x)
g ( x)
Piecewise-Defined Function



A piecewise-defined function is a function that is
defined by two or more equations over a specified
domain.
The absolute value function f x   x
can be written as a piecewise-defined function.
The basic characteristics of the absolute value
function are summarized on the next page.
Example

Evaluate the function when x = -1 and 0.
Domain of a Function

The domain of a function can be implied by
the expression used to define the function

The implied domain is the set of all real
numbers for which the expression is defined.

For example,

The function
has an implied
domain that consists of all real x other than
x = ±2

The domain excludes x-values that result
in division by zero.


Another common type of implied domain is
that used to avoid even roots of negative
numbers.
EX:
is defined only for x  0.
The domain excludes x-values that result
in even roots of negative numbers.