Relations and functions

A relation is a set of ordered
pairs.
domain = {-1,0,2,4,9}
These are the x values written in a set from smallest to largest
{(2,3), (-1,5), (4,-2), (9,9), (0,-6)}
This is a
relation
The domain is the set of all x values in the relation
These are the y values written in a set from smallest to largest
range = {-6,-2,3,5,9}
The range is the set of all y values in the relation

A relation assigns the x’s with y’s
1
2
3
4
5
2
4
6
8
10
Domain (set of all x’s) Range (set of all y’s)
This relation can be written {(1,6), (2,2), (3,4), (4,8), (5,10)}

A function f from set A to set B is a rule of correspondence
that assigns to each element x in the set A exactly element y in the set B.
A function f from set A to set B is a rule of correspondence
that assigns to each element x in the set A exactly one
element y in the set B.
1
2
3
4
5
Set A is the domain
Whew! What
did that say?
2
4
6
8
10
Set B is the range
Must use all the x’s
The x value can only be assigned to one y
This is a function
---it meets our
conditions

Let’s look at another relation and decide if it is a function.
The second condition says each x can have only one y, but it CAN
be the same y as another x gets assigned to.
1
2
3
4
5
Set A is the domain
2
4
6
8
10
Set B is the range
This is a function
---it meets our
conditions

A good example that you can “relate” to is students in our
maths class this semester are set A. The grade they earn out
of the class is set B. Each student must be assigned a grade
and can only be assigned ONE grade, but more than one
student can get the same grade (we hope so---we want lots
of A’s). The example show on the previous screen had each
student getting the same grade. That’s okay.
A good example that you can “relate” to is students in our
maths class this semester are set A. The grade they earn out
of the class is set B. Each student must be assigned a grade
and can only be ONE grade, but more than one
student can get the same grade (we hope so---we want lots of
A’s). The example shown on the previous screen had each
student getting the same grade. That’s okay.
1
2
3
4
5
2
4
6
8
10
2 was assigned both 4 and 10
Is the relation shown above a function? NO Why not???

Check this relation out to determine if it is a function.
It is not---3 didn’t get assigned to anything
Comparing to our example, a student in maths must receive a grade
1
2
3
4
5
Set A is the domain
2
4
6
8
10
Set B is the range
This is not a
function---it
doesn’t assign
each x with a y

Check this relation out to determine if it is a function.
This is fine—each student gets only one grade. More than one can
get an A and I don’t have to give any D’s (so all y’s don’t need to be
used).
1
2
3
4
5
Set A is the domain
2
4
6
8
10
Set B is the range
This is a function

We commonly call functions by letters. Because function
starts with f, it is a commonly used letter to refer to
functions.
  2 3 6 2 f x  x  x 
This means
the right
hand side is
a function
called f
This means
the right hand
side has the
variable x in it
The left side DOES NOT
MEAN f times x like
brackets usually do, it
simply tells us what is on
the right hand side.
The left hand side of this equation is the function notation.
It tells us two things. We called the function f and the
variable in the function is x.

  2 3 6 2 f x  x  x 
2 22 32 6 2 f   
Remember---this tells you what
is on the right hand side---it is
not something you work. It says
that the right hand side is the
function f and it has x in it.
f 2  24 32 6  8  6  6  8
So we have a function called f that has the variable x in it.
Using function notation we could then ask the following:
Find f (2).
This means to find the function f and instead of
having an x in it, put a 2 in it. So let’s take the
function above and make brackets everywhere
the x was and in its place, put in a 2.
Don’t forget order of operations---powers, then
multiplication, finally addition & subtraction

  2 3 6 2 Find f (-2). f x  x  x 
 2 2 2 3 2 6 2 f      
f  2  243 2 6  8 6 6  20
This means to find the function f and instead of having an x
in it, put a -2 in it. So let’s take the function above and make
brackets everywhere the x was and in its place, put in a -2.

  2 3 6 2 Find f (k). f x  x  x 
  2  3  6 2 f k  k  k 
  2  3  6 2 3 6 2 2 f k  k  k   k  k 
This means to find the function f and instead of having an x
in it, put a k in it. So let’s take the function above and make
brackets everywhere the x was and in its place, put in a k.

  2 3 6 2 Find f (2k). f x  x  x 
2  22  32  6 2 f k  k  k 
2  24  32  6 8 6 6 2 2 f k  k  k   k  k 
This means to find the function f and instead of having an x in
it, put a 2k in it. So let’s take the function above and make
brackets everywhere the x was and in its place, put in a 2k.

Let's try a new function
gx x 2x 2  
Find g(1)+ g(-4).
1 1 21 1 2 g    
 4  4 2 4 16 8 24 2 g        
So g1 g 4  1 24  23

The last thing we need to learn about functions for
this section is something about their domain. Recall
domain meant "Set A" which is the set of values you
plug in for x.
For the functions we will be dealing with, there
are two "illegals":
1. You can't divide by zero (denominator (bottom)
of a fraction can't be zero)
2. You can't take the square root (or even root) of
a negative number
When you are asked to find the domain of a function,
you can use any value for x as long as the value
won't create an "illegal" situation.

Find the domain for the following functions:
f x  2x 1
Since no matter what value you
choose for x, you won't be dividing
by zero or square rooting a negative
number, you can use anything you
want so we say the answer is:
All real numbers x.
Note: There is
nothing wrong with
the top = 0 just means
the fraction = 0
 

3

2

x
x
g x
If you choose x = 2, the denominator
will be 2 – 2 = 0 which is illegal
because you can't divide by zero.
The answer then is:
All real numbers x such that x ≠ 2.
means does not equal
illegal if this
is zero

Let's find the domain of another one:
hx  x  4
Can't be negative so must be ≥ 0
0 4   x solve
this 4  x
We have to be careful what x's we use so that the second
"illegal" of square rooting a negative doesn't happen. This
means the "stuff" under the square root must be greater
than or equal to zero (maths way of saying "not negative").
So the answer is:
All real numbers x such that x ≠ 4

Summary of How to Find the
Domain of a Function
• Look for any fractions or square roots that could cause one
of the two "illegals" to happen. If there aren't any, then the
domain is All real numbers x.
• If there are fractions, figure out what values would make the
bottom equal zero and those are the values you can't use.
The answer would be: All real numbers x such that x ≠
those values.
• If there is a square root, the "stuff" under the square root
cannot be negative so set the stuff ≥ 0 and solve. Then
answer would be: All real numbers x such that x ≠
whatever you got when you solved.
NOTE: Of course your variable doesn't have to be x, can be
whatever is in the problem.

Acknowledgement
I wish to thank Shawna Haider from Salt Lake Community College, Utah
USA for her hard work in creating this PowerPoint.
www.slcc.edu
Shawna has kindly given permission for this resource to be downloaded
from www.mathxtc.com and for it to be modified to suit the Western
Australian Mathematics Curriculum.
Stephen Corcoran
Head of Mathematics
St Stephen’s School – Carramar
www.ststephens.wa.edu.au

Relations and functions

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