Functions
ƒ(x) Function Notations
A relation is a pairing between two sets. A
function is a relation in which each x-value has
only one y-value
Functions can be represented in many ways including
tables, graphs and equations.
Take for example the equation y = 2x - 3. This
equation has an important characteristic. For
each value of x, you find exactly one value of y.
Relation:
A relation is simply a set of
ordered pairs.
A relation can be any set of
ordered pairs. No special
rules need apply.
The following is an example of
a relation:
{(1,2)(2,4)(3,5)(2,6)(1,-3)}
The graph at the right shows
that a vertical line may
intersect more than one point
in a relation.
Function: A function is a set
of ordered pairs in which
each x-value has only ONE yvalue associated with it.
The relation we just
discussed
{(1,2),(2,4,)(3,5)(2,6)(1,-3)}
is NOT a function because
the x-value 2 is paired with a
y-value of 4 and 6. Similarly,
the x-value of 1 is paired with
the y-value of 2 and -3
The previous relation can
be altered to become a
function by removing the
ordered pairs where the xvalue is used twice.
Function: {(1,2)(2,4)(3,5)}
The graph at the left
shows that a vertical line
intersects only ONE point
in a function. This is called
the vertical line test for
functions.
Function – an input-output relationship that has
exactly one output for each input.
Domain – the set of all input (x)values of a
function.
Range – The set of all output (y)values in a
function.
Function notation – the notation used to describe
a function.
Example f(x) is read “f of x.”
f(1) is read “f of 1.”
Linear function – a function whose graph is a
straight line.
To determine if a relationship is a function,
verify that each input has exactly one output.
Using tables is one way to verify functions.
Look at the function below. Can you determine
if it is a relation or a function?
You can identify functions using tables or
graphs. The graph below has more than one
output for each input. Is this a function?
A relation can be represented by a set of
order pairs (x, y) . The first number, x, is a
member of the domain and the second number,
y, is a member of the range.
Determine whether the order pairs make a function.
{(-1, 7), (0, 3), (1, 5), (0, -3)}
{(0, 2), (2, 4), (4, 8), (8, 10)}
Vertical-Line Test for a function
If no vertical line in the coordinate plane
intersects a graph in more than one point, then
the graph represents a function.
(You can use a pencil held vertically to test)
Evaluating Functions
For the function y = 2x - 1, find f(0), f(2), and f(-1).
y = 2x – 1
f(x) = 2x -1 Write in function notation.
f(0) = 2(0) – 1 = -1
f(2) = 2(2) – 1 = 3
f(-1) = 2(-1) – 1 = -3
Find f(1), f(2), f(3), and f(4).
Read the graph to find y for each x.
f(x) = y
f(1) = 8
f(2) = 10
f(3) = 12
f(4) = 14
Linear Functions
Straight Lines
Writing equations of functions
Use the equation f(x) = mx + b.
Find b (y-intercept) = -4
Locate a point on the line, such as
(2, 0).
Substitute the values into your
equation.
f(x) = mx + b
0 = m(2) – 4
0 = 2m – 4
0 + 4 = 2m -4 + 4
4 = 2m
2 2
m=2
f(x) = 2x - 4
Writing an equation using a table
X
Y
-2
-3
-1
-1
0
1
1
3
2
5
The y-intercept can be identified from
the table, (0, 1)
Pick a point, (1, 3) and substitute your
point and y-intercept into your equation.
f(x) = mx + b
3 = m(1) + 1
3=m+1
3–1=m+1–1
m=2
f(x) = 2x + 1
Physical Science
The relationship between the two temperatures in
the table are linear.
Write a rule for Fahrenheit temperature as a
function of Celsius temperature.
f(x) = mx + b, where x is Celsius and y is Fahrenheit
Temperature (°C)
-10
0
25
50
100
Temperature (°F)
14
32
77
122
212
Practice with Functions
1)
Which of the relations below is a function?
a) {(2, 3), (3, 4), (5, 1), (2, 4)}
b) {(2, 3), (3, 4), (6, 2), (7, 3)}
c) {(2, 3), ( 3, 4), (6, 2), (3, 3)}
2) Given the relation A = {(5, 2), (7, 4), (9, 10), (x, 5)}.
Which of the following values for x will make
relation A a function?
a) 7
b) 9
c) 4
3)
The following relation is a function.
{(10, 12), (5, 3), (15, 10), (5, 6), (1, 0)}
• True
• False
4) Which of the relations below is a function?
a)
b)
c)
{(1, 1,), (2, 1), (3, 1), (4, 1), (5, 1)}
{(2, 1), (2, 2), (2, 3), (2, 4), (2, 5)}
{(0, 2), (0, 3), (0, 4), (0, 5), (0, 6)}
5)
The graph of a
relation is shown at
the right. Is this
relation a function?
a) yes
b) no
c) Cannot be
determine from a
graph
6) Is the relation depicted in the table below
a function?
a) yes
b) no
c) cannot be determined from a table
X
0
1
3
5
3
9
Y
8
9
10
6
10
7
7)
The graph of a relation is shown below.
Is the relation a function?
a) yes
b) no
c) cannot be determined from a graph
8) Is the relation in the table below a
function?
a) yes
b) no
x
-2
-1
0
1
2
3
y
5
5
5
5
5
5
9)
The graph of a relation is shown below.
Is the relation a function?
a) yes
b) no
c) cannot be determined from a graph
10) The graph of a relation is shown below. Is
the relation a function?
a) yes
b) no
c) cannot be determined from a graph
11) Given f(x) = 3x + 7, find f(5).
a)
b)
c)
15
22
42
12)
Which graph represents a function?