SECTION 16.1: 5 WAYS TO
CONCEPTUALIZE A FUNCTION
In an earlier section, you were introduced to a very basic definition of a
function. I stated that a function is a set of ordered pairs such that any
given xvalue in the set of ordered pair has only one unique yvalue assigned
to it. Below are 5 different ways to model the concept of a function.
SECTION 16.2: FUNCTIONS AND RESEARCH
Researchers often collect data as ordered pairs. Functions tend to be
the "essence" of "real world" problems. What does this previous
statement mean?
Researchers tend to explore situations where "function" relationships occur.
For example, at a given instant in time, you have one weight. I'm sure at the
time you wake up in the morning, say 6 a.m., you weigh a fixed amount, say 165
pounds. I don't think you could ever imagine an instance where you might weigh
165 pounds and 184 pounds at the same time 6 a.m. on the same morning.
How about this? What if someone said to you: "At 11:30 a.m. today, Mr. Landry
will be in two different places at Deering High School." I don't think you
would believe it, because you believe in the "functional nature" of our
everyday experience. At a particular time t you know that I will be at a
particular position P: that's our everyday experience that we depend on. Such
accepted human perceptions on the "nature of the world" are the "essence" of
the mathematical definition of a function.
Below is a table of ordered pairs collected by census collectors in Mexico
over a sevenyear period beginning in 1980 and a graph of these ordered pairs
to the right of the table.
Of course the above graph of the research data is a function. But is there a
quick way to determine a graph is a function without looking through all the
ordered pairs? \
SECTION 16.3: VERTICAL LINE
TEST TO DETERMINE IF A GRAPH IS A FUNCTION
If you have a graph of ordered pairs and you want to have a quick test to see
if the graph is a function, place a vertical line over the graph. If the
vertical line touches only one point of the graph no matter where you move
that vertical line then you have a function. Consider the graphs of
the same function below:
As you can see, as I move the vertical line from left to right it will only
touch one point at any time, which implies that the t value does not have two
different p values. This test for determining if a graph is a function
is called the "VERTICAL LINE TEST."
SECTION 16.4: GRAPHS OF
ORDERED PAIRS THAT ARE NOT FUNCTIONS
Examine the graph below. The set of ordered pairs on the graph below are { (1,0), (0, 1), (1, 0), (0, 1) }.
If you use the vertical line test on the graph above, you will see that a
vertical line will go through two points when placed on the graph.
Look below.
This vertical line above goes through two points (0, 1) and (0, 1). Can you
see that we have a situation where an xvalue, in this instance, x = 0, has
two different yvalues, y = 1 and y = 1. So we know that this graph is not a
function from the Vertical Line Test. But we already knew
this because we had the set of ordered pairs { (1,0), (0, 1), (1, 0), (0, 1)
} that showed us that an xvalue had two different yvalues.
SECTION 16.5: INVERSE OF A
FUNCTION AND THE HORIZONTAL LINE TEST
If you have a function and swap the x and y values of each ordered pair of the
function, and the resulting set of ordered pairs is a
function, that resulting set of ordered pairs is
called the "inverse" of the
function.
Below is the table of values from a previous problem. The ordered pairs are {
(0, 7), (1, 10), (2, 13),...,(5, 22) } If you swapped the t and d values you
would get the set of ordered pairs { (7, 0), (10, 1), (13, 2),...,(22, 5) }.
This resulting set of ordered pairs is a function so this resulting set of
ordered pairs would be called the "inverse" of the function {
(7, 0), (10, 1), (13, 2),...,(22, 5) }.
Here is another example of a function:
.
Notice that each xvalue has one and only one yvalue. Now let's swap the
ordered pairs in the function
.
We would get:
.
Notice that the resulting ordered pairs is NOT a function
because the xvalue 5 has two different yvalues, 2 and 7. Therefore, we would
say that the function
does not have an inverse.
IMPORTANT

A function DOES NOT necessarily have an inverse.

Examples: Which of the functions have an inverse?
a)
Swap the ordered pairs:
 This is a function.
Therefore
has an inverse and it is .
b)
Swap the ordered pairs:
 This is not a function.
Since
and
are two xvalues that are the same and have different yvalues we say
does not have an inverse.
HORIZONTAL LINE TEST FOR DETERMINING
IF A FUNCTION HAS AN INVERSE

If you are given a function as a graph then there is a quick way to determine if it has an inverse is to use the "HORIZONTAL LINE TEST."
HORIZONTAL LINE TEST
"if a horizontal line goes through more than one point on a function then the function does not have an inverse.
Below is the graph of .
Notice that the horizontal line y = 2 goes through the two points
(1,1) and (1,1) which are on .
Therefore y =
x does not have an inverse since (1, 1) and (1, 1)
would become (1, 1) and (1, 1) on the "inverse" which would produce two
xvalues that are the same but would have different yvalues.
REMEMBER AN INVERSE MUST BE A FUNCTION 

SECTION 16.6: FUNCTION NOTATION
Some equations produce ordered pairs that are functions. If you produced
ordered pairs from y = 4x  3, you would get a set of ordered pairs that would
fall on a straight line tilted to the right. Why? Because this is an equation
in the form of y = mx + b which produces ordered pairs that follow the path of
a line. And since the slope, m = 4, is positive the line tilts to the right.
If an equation is known to be a function,
which y = 4x  3 is, mathematicians use an alternative symbolism for
the variable y or any other variable you so happen to use on the lefthand
side of the equation that is a function. Compare the following two equations:
Notice the circled algebraic expressions above? Notice that each of those
circled expressions are the same? Notice the symbols on the left hand side of
these two equations are different? One equation contains the familiar y, while
the other equation contains the lower case letter "f" followed by x in
parentheses. Compare the following equations again:
Notice that the circled mathematical expressions above are the same? Notice
that the symbols on the lefthand side of the equations are different? One
equation contains the familiar y, while the second equation contains the lower
case letter "f" followed by x in parentheses; the third, contains the lower
case letter "w" followed by x in parentheses; the fourth, contains the lower
case letter "h" followed by x in parentheses.
The symbols f(x), w(x), and h(x), above, all start with a lower case letter of
the alphabet followed by parentheses that contain the same variable
that is on the right hand side of each of the above equations.
These symbols are examples of "function notation." If you see
this type of symbol on the left hand side of an equation it means that the
expression on the right hand side of the equation will produce ordered pairs
that will in fact be a function.
So, if I gave you the following equation, ,
then I am letting you know that the algebraic expression
will produce a set of ordered pairs that will be a function.

"Function Notation" is used to let you know
that the algebraic expression it is
assigned to will produce a set of ordered
pairs that is a function.

SECTION 16.7: INCORRECT USE OF
FUNCTION NOTATION
In
the mathematical expression on the righthand side contains the variable t and
so does the function notation s(t). Notice that there is a t in the
parentheses of s(t). I said that this must occur if you use function notation
correctly. You would never see expressions like
or
in mathematics.
SECTION 16.8: FUNCTION
NOTATION AS A DELIVERER OF INFORMATION FOR PRODUCING ORDERED PAIRS
Consider the following function
and the following notation
and
Whether you realize it or not, you have 5 different sets of instructions!
Let's deal with the symbol
first.
The symbol
""
is a set of instructions. Here are the instructions:

First, you are being asked to take the number inside of the parentheses of
s(0), in this case 0, and assign it to t, the variable in the algebraic
expression .
That is, let t = 0.

Second, you are being asked to take that number assigned to t, i.e. 0, and
then substitute it into the algebraic expression that s(t) has been set equal
to, in this case
That is,

Third, evaluate the algebraic expression, i.e. .

Fourth, you assign the function notaion to the result of your evaluation, i.e. .

Fifth, create an ordered pair using the value of t, i.e. 0, and the resulting
evaluation, i.e. 5. That is, (0, 5)
You will use the same principle above to perform the following instructions:
s(1), s(4), s(7), and s(5).
Then your ordered pair will be (1, 1).
Then your ordered pair will be (4, 11).
Then your ordered pair will be (7, 121).
Then your ordered pair will be (5, 25).
Interesting use of symbolism? On the next page let's compare and contrast xy
symbolism with function notation symbolism.
SECTION 16.9: COMPARING AND
CONTRASTING "y =" SYMBOLISM WITH FUNCTION NOTATION SYMBOLISM
In the past you have seen equations like
but not like .
The only difference between these two equations is that one uses y and the
other s(t). To create ordered pairs with
you would choose an arbitrary number for t, substitute that number into the
mathematical expression ,
and then assign the result of your evaluation to y. See below:
Using y =
notation: :
.
Substitute t = 3 into
to get .
So when t = 3 the result for y is 25.
We assign y = 25..
(3, 25) is the resulting ordered pair.
Using function notation: Find s(3).
t = 3:
substitute 3 into
to get
So when t = 3 the result for is 25. We assign
(3, 25) is the resulting ordered pair.
SECTION 16.10: INDEPENDENT AND
DEPENDENT VARIABLES IN SCIENCE
In order for you to stay alive, you are "dependent" on air,
water and food. Without these three things you cannot live. Your existence is
"dependent" on some other things.
As you know the word "variable" means change. Your existence
is variable because you change throughout your life.
We are dependent and we are variable: we are in fact a "dependent
variable". As a "dependent variable" we are
dependent on other things like food, water, and air, and these things change
us through time.
The things that we depend on for our existence could be called
"independent variables." See the chart below:

AIR > INDEPENDENT VARIABLE
YOU> DEPENDENT VARIABLE
WATER > INDEPENDENT VARIABLE
YOU > DEPENDENT VARIABLE
FOOD > INDEPENDENT VARIABLE
YOU > DEPENDENT VARIABLE
HYDROGEN > INDEPENDENT VARIABLE
WATER(H_{2}O) > DEPENDENT VARIABLE
OXYGEN > INDEPENDENT VARIABLE
WATER(H_{2}O) > DEPENDENT VARIABLE

Researchers in the sciences are
always searching out relationships between independent and dependent
variables. Why? Because that understanding is the basis for us learning how to
control our environment and how our environment controls us.
SECTION 16.11: INDEPENDENT AND
DEPENDENT VARIABLES IN MATHEMATICS
When you have an equation like y = 3x + 5 or any equation for that matter, the variables x and y have special names
assigned to them. Before you can get a yvalue you need to have an xvalue. In
other words, the result assigned to y is dependent on the value assigned to x.
You cannot get a yvalue without an xvalue.
Given what we know about independent and dependent variables in science, it
does not seem unreasonable to say that x is the independent variable and y is
the dependent variable in the equation y = 3x + 5.
You can't get a value for y without first being given a value for x. The value
of y is in fact dependent on the value assigned to x.
If we used function notation like f(x) = 3x + 5 we would say x is the
independent variable and f(x) is the dependent variable.
Examples) Name the independent and dependent variables in:

independent variable 


dependent variable 
a) 
x 


s 
b) 
f 


u(f) 
c) 
d 


g 
d) 
e 


d(e) 
SECTION 16.12: HOW TO SPEAK
FUNCTION NOTATION
Now it seems reasonable after doing all this work with functions that you
should learn how to pronounce something like f(x), g(x), q(x), or any function
notation for that matter.

Function Notation 


You Say 
a) 
f(x) 


f of x 
b) 
g(x) 


g of x 
c) 
r(x) 


r of x 
d) 
m(x) 


m of x 
SECTION 16.13: "IN TERMS OF..."
Another phrase you'll have to get use to when dealing with function notation
is the phrase "in terms of." Let me give a few examples of
this new language, and then we will pick that language apart.
ex. We
would say that the function g is in terms of x.
ex.
We would say that the function m is in terms of u.
