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Functions
A relation is a set of ordered
pairs.
The domain is the set of all x values in the relation
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
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
2
4
6
5
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)}
functionf ffrom
fromset
setXXXinto
into
Let X and Y be two nonempty sets. A function
from
set
into
set Y is a rule of correspondence (relation) that associates with
each element
each
elementxxininthe
theset
setXXexactly
exactly
exactlyone
oneelement
elementyyyin
inthe
theset
setY.
Y.
one
element
in
the
set
Y.
1
2
3
4
5
2
4
6
8
10
Set X is the domain
What
This is a Whew!
function
did that
---it meets
oursay?
conditions
Set Y is the range
Must use all the x’s
The x value can only be assigned to one y
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
2
4
6
8
10
Set X is the domain
This is a function
---it meets our
conditions
Set Y is the range
Must use all the x’s
The x value can only be assigned to one y
A good example that you can “relate” to is students in College
Algebra this semester are Set X. The grade they earn out of
the class is Set Y. Each student must be assigned a grade
and can
can only
onlybe
beassigned
assignedONE
ONEgrade,
grade,
but
but
more
more
than
than
one
one
student can get the same grade (we hope so---we want lots of
A’s).
of
A’s).
The
The
example
example
shown
shown
onon
thethe
previous
previous
screen
screen
had
had
each
each
student getting the same grade. That’s okay.
1
2
3
4
5
A
B
C
D
E
Student 2 was assigned both B and E
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 College Algebra must
receive a grade
1
A
2
B
3
C
4
D
E
5
Set X is the domain
Set Y is the range
This is not a
Must use all the x’s
function---it
doesn’t assign
each x with a y The x value can only be assigned to one 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
A
B
C
D
E
Set X is the domain
This is a function
Set Y is the range
Must use all the x’s
The x value can only be assigned to one y
We commonly call functions by letters. Because
function starts with f, it is a commonly used letter to
refer to functions.
f  x   2 x  3x  6
2
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
parenthesis 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.
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  x   2 x  3x  6
2
f 2   22   32   6
2
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:
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
Find f (2).
function above and make parenthesis everywhere
the x was and in its place, put in a 2.
Don’t forget order of operations---powers, then
multiplication, finally addition & subtraction
Find f (-2).
f  x   2 x  3x  6
2
f  2  2 2  3 2  6
2
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
parenthesis everywhere the x was and in its place, put in a -2.
Don’t forget order of operations---powers, then
multiplication, finally addition & subtraction
f  x   2 x  3x  6
2
Find f (k).
f k   2k   3k   6
2
 
f k   2 k  3k   6  2k  3k  6
2
2
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
parenthesis everywhere the x was and in its place, put in a k.
Don’t forget order of operations---powers, then
multiplication, finally addition & subtraction
f  x   2 x  3x  6
2
Find f (2k).
f 2k   22k   32k   6
2
 
f 2k   2 4k  32k   6  8k  6k  6
2
2
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
parenthesis everywhere the x was and in its place, put in a 2k.
Don’t forget order of operations---powers, then
multiplication, finally addition & subtraction
Let's try a new function
Find g(1)+ g(-4).
g x   x  2 x
2
g 1  1  21  1
2
g  4   4  2 4  16  8  24
2
So g 1  g  4  1  24  23
Another thing we need to learn about functions for
this section is something about their domain. Recall
domain meant "Set X" 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 (we are talking real numbers)
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:
Since no matter what value you
choose for x, you won't be dividing
f x  2x 1 by zero or square rooting a negative
number, you can use anything you
Note: There is
want so we say the answer is:
nothing wrong with
the top = 0 just means All real numbers x.

the fraction = 0
x3
g x  
x2
illegal if this
is zero
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
Let's find the domain of another one:
h x   x  4
Can't be negative so must be  0
x4 0
solve
this
x4
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 (math 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; it can be
whatever is in the problem.
We’ve already graphed equations. We can graph functions
in the same way. The thing to remember is that on the graph
the f(x) or function value is the same as the y value.
If we want to graph the function f (x) = 3x – 1, it is
the same thing as graphing y = 3x – 1.
We recognize this as a line with a slope 3 and y-intercept of -1
So to graph any functions given,
simply write a y where you see f(x)
and then graph with the same method
as you did graphs of equations
plugging in values for x and finding the
corresponding y values and plotting
the points. Also recall that the domain
is the x-values you can “legally” plug
in and the range is the y-values you
get out.
The other thing you need to know is
how to tell from a graph if the graph is
of a function or not. This is addressed
on the next screen
Recall that for a relation to be a function, for each x there can
only be one y value. Let’s look at a couple of graphs.
yx
2
x y
2
x=0 x=1
x = x-2= -1
x =x 0= 1
Look at different x values
and see there is only one
y value on the graph for it.
This is a function
At x = 1 there are two y
values.
This then is NOT a
function
From what we've just seen, we can tell by looking at a
graph of an equation if it is a function or not by what
we call the vertical line test.
If a vertical line intersects the graph of an
equation more than one time, the equation
graphed is NOT a function.
This is a function
This is NOT a
function
This is a function
y


Is the graph shown a function?
(0, 2)











x

Yes, a vertical line would only intercept
the graph one time so for every x in the
domain, there is exactly one y.



x = 0 here
What is f(0)?
f(0) = 2
This is asking you to find the function
value (the y value) when the x is 0.
What is the domain of the function?
Since there are not any places where x is not defined on this
graph, the domain is all real numbers.
What is the range of the function?
The range is the y values. This graph goes from negative to
positive infinity so the range is all real numbers.
from neg  to
1 it is above
the axis so is
greater than 0



y





Is the graph shown a function?
(1, 2)
Yes, a vertical line would only intercept
the graph one time so for every x in the
from 0 to  domain, there is exactly one y.






x

it is above
the axis so
is greater
than 0
What is f(1)?
f(1) = 2
This is asking you to find the function
value (the y value) when the x is 1.
x = 1 here
What is the domain of the function?
The graph doesn't exist when x = 0 x x  0




What is the range of the function?
The range is the y values. Notice that y does not ever equal 1.
The range then is: y y  1


For what values of x is f(x) > 0?
(, 1) and (0, )
Look at the graph and see where f(x) (the y value) is greater
than 0.