We now need to
move into the second topic of this chapter. The first thing that we need to do is
define just what a function is. There are lots and lots of definitions for a
function out there and most of them involve the terms rule, relation,
or correspondence. While these are more technically accurate than the
definition that we’re going to use in this section all the fancy words used in
the other definitions tend to just confuse the issue and make it difficult to understand
just what a function is.
So, here is the
definition of function that we’re going to use. Again, I need to point out that
this is NOT the most technically accurate definition of a function, but it is a
good “working definition” of a function that helps us to understand just how a
function works.
“Working
Definition” of Function
A function is
an equation (this is where most definitions use one of the words given above)
if any x that can be plugged into the equation will yield exactly one y
out of the equation.
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There it is.
That is the definition of functions that we’re going to use. Before we examine
this a little more note that we used the phrase “x that can be plugged
into” in the definition. This tends to imply that not all x’s can be
plugged into and equation and this is in fact correct. We will come back and
discuss this in more detail towards the end of this section, however at this
point just remember that we can’t divide by zero and if we want real numbers
out of the equation we can’t take the square root of a negative number. So,
with these two examples it is clear that we will not always be able to plug in
every x into any equation.
When dealing
with functions we are always going to assume that both x and y will
be real numbers. In
other words, we are going to forget that we know anything about complex numbers for a little bit
while we deal with this section.
Okay, with that
out of the way let’s get back to the definition of a function. Now, we started
off by saying that we weren’t going to make the definition confusing. However,
what we should have said was
we’ll try not to make it too confusing, because no matter how we define it the definition
is always going to be a little confusing at first.
We
now need to move onto something called function notation. Function
notation will be used heavily throughout most of the remaining chapters in this
course and so it is important to understand
it.
Let’s
start off with the following quadratic equation.
y = x2 − 5x + 3
We
can use a process similar to what we used in the previous set of examples to
convince ourselves
that this is a function. Since this is a function we will denote it as follows,
f (x) = x2 − 5x + 3
So,
we replaced the y with the notation f ( x) . This is read
as “f of x”. Note that there is nothing special
about the f we used here. We could just have easily used any of the
following,
g (x) = x2 − 5x + 3 h(x) = x2 − 5x + 3 R(x) = x2 − 5x + 3
The
letter we use does not matter. What is important is the “ (x) ” part. The
letter in the parenthesis
must match the variable used on the right side of the equal sign.
It
is very important to note that f ( x) is really
nothing more than a really fancy way of writing y. If you keep that in
mind you may find that dealing with function notation becomes a little easier.
Also,
this is NOT a multiplication of f by x! This is one of the
more common mistakes people make
when the first deal with functions. This is just a notation used to denote
functions.
Next
we need to talk about evaluating functions. Evaluating function is
really nothing more than
asking what its value is for specific values of x. Another way of
looking at it is that we are asking
what the y value for a given x is.
Evaluation
is really quite simple. Let’s take the function we were looking at above
f (x) = x2 − 5x + 3
and
ask what its value is for x = 4 . In terms of function notation we will “ask” this using the notation
f (4) . So, when there
is something other than the variable inside the parenthesis we are really
asking what the value of the function is for that particular quantity.
Now,
when we say the value of the function we are really asking what the value of
the equation is for that particular value of x. Here is f (4) .
f (4) = (4)2 − 5 (4) + 3 = 16 − 20 + 3 = −1
Notice
that evaluating a function is done in exactly the same way in which we evaluate
equations. All we do is plug in for x whatever is on the inside of the
parenthesis on the left. Here’s another evaluation for this function.
f (−6) = (−6)2 − 5 (−6) + 3 = 36 + 30 + 3 = 69
So,
again, whatever is on the inside of the parenthesis on the left is plugged in
for x in the equation
on the right.
Function
evaluation is something that we’ll be doing a lot of in later sections and
chapters so make sure that you can do it. You will find several later sections
very difficult to understand and/or do the work in if you do not have a good
grasp on how function evaluation works.
While
we are on the subject of function evaluation we should now talk about piecewise
functions. We’ve actually already seen an example of a piecewise function
even if we didn’t call it a function (or a piecewise function) at the
time. Recall the mathematical definition of absolute value.
This
is a function and if we use function notation we can write it as follows,
This
is also an example of a piecewise function. A piecewise function is nothing
more than a function that is broken into pieces and which piece you use depends
upon value of x. So, in the absolute value example we will use the top
piece if x is positive or zero and we will use the bottom piece if x is
negative.
Piecewise
functions do not arise all that often in an Algebra class however, the do arise
in several places in later classes and so it is important for you to understand
them if you are going to be moving on to more math classes.
As a
final topic we need to come back and touch on the fact that we can’t always
plug every x into every function. We talked briefly about this when we
gave the definition of the function and we saw an example of this when we were
evaluating functions. We now need to look at this in a little more
detail.
First
we need to get a couple of definitions out of the way.
Domain
and Range
The
domain of an equation is the set of all x’s that we can plug
into the equation and get back a real number for y. The range of
an equation is the set of all y’s that we can ever get out of the equation.
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Note
that we did mean to use equation in the definitions above instead of functions.
These are really
definitions for equations. However, since functions are also equations we can
use the definitions for functions as well.
Determining
the range of an equation/function is can be pretty difficult to do for many
functions and
so we aren’t going to really get into that. We are much more interested here in
determining the domains of functions. From the definition the domain is the set
of all x’s that we can plug into a function and get back a real number.
At this point, that means that we need to avoid division by zero and taking
square roots of negative numbers.
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