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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
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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.
In order to clear up some of the confusion let’s look at
some examples of equations that are functions and equations that aren’t
functions.
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Example 1 Determine
which of the following equations are functions and which are not functions.
(a)  [Solution]
(b) [Solution]
(c) [Solution]
(d) [Solution]
Solution
The definition of function is saying is that if we take
all possible values of x and plug
them into the equation and solve for y
we will get exactly one value for each value of x. At this stage of the
game it can be pretty difficult to actually show that an equation is a
function so we’ll mostly talk our way through it. On the other hand it’s often quite easy to
show that an equation isn’t a function.
(a)
So, we need to show that no matter what x we plug into the equation and solve
for y we will only get a single
value of y. Note as well that the value of y will probably be different for each
value of x, although it doesn’t
have to be.
Let’s start this off by plugging in some values of x and see what happens.
So, for each of these value of x we got a single value of y
out of the equation. Now, this isn’t
sufficient to claim that this is a function.
In order to officially prove that this is a function we need to show
that this will work no matter which value of x we plug into the equation.
Of course we can’t plug all possible value of x into the equation. That just isn’t physically possible. However, let’s go back and look at the ones
that we did plug in. For each x, upon plugging in, we first
multiplied the x by 5 and then
added 1 onto it. Now, if we multiply a
number by 5 we will get a single value from the multiplication. Likewise, we will only get a single value
if we add 1 onto a number. Therefore,
it seems plausible that based on the operations involved with plugging x into the equation that we will only
get a single value of y out of the
equation.
So, this equation is a function.
[Return to Problems]
(b)
Again, let’s plug in a couple of values of x and solve for y to see what happens.
Now, let’s think a little bit about what we were doing
with the evaluations. First we squared
the value of x that we plugged
in. When we square a number there will
only be one possible value. We then
add 1 onto this, but again, this will yield a single value.
So, it seems like this equation is also a function.
Note that it is okay to get the same y value for different x’s. For example,
We just can’t get more than one y out of the equation after we plug in the x.
[Return to Problems]
(c)
As we’ve done with the previous two equations let’s plug
in a couple of value of x, solve
for y and see what we get.
Now, remember that we’re solving for y and so that means that in the first and last case above we will
actually get two different y values
out of the x and so this equation
is NOT a function.
Note that we can have values of x that will yield a single y
as we’ve seen above, but that doesn’t matter.
If even one value of x
yields more than one value of y
upon solving the equation will not be a function.
What this really means is that we didn’t need to go any
farther than the first evaluation, since that gave multiple values of y.
[Return to Problems]
(d)
With this case we’ll use the lesson learned in the
previous part and see if we can find a value of x that will give more than one value of y upon solving. Because
we’ve got a y2 in the
problem this shouldn’t be too hard to do since solving will eventually mean
using the square root property which
will give more than one value of y.
So, this equation is not a function. Recall, that from the previous section this
is the equation of a circle. Circles
are never functions.
[Return to Problems]
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Hopefully these examples have given you a better feel for
what a function actually is.
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.
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,
So, we replaced the y
with the notation . 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,
The letter we use does not matter. What is important is the “ ” 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 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
and ask what its value is for . In terms of function notation we will “ask”
this using the notation . 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 .
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.
So, again, whatever is on the inside of the parenthesis on
the left is plugged in for x in the
equation on the right. Let’s take a look
at some more examples.
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Example 2 Given
and evaluate each of the following.
(a) and [Solution]
(b) and [Solution]
(c) [Solution]
(d) [Solution]
(e) and [Solution]
(f) [Solution]
(g) [Solution]
Solution
(a) and
Okay we’ve got two function evaluations to do here and
we’ve also got two functions so we’re going to need to decide which function
to use for the evaluations. The key
here is to notice the letter that is in front of the parenthesis. For we will use the function and for we will use . In other words, we just need to make sure
that the variables match up.
Here are the evaluations for this part.
[Return to Problems]
(b) and
This one is pretty much the same as the previous part with
one exception that we’ll touch on when we reach that point. Here are the evaluations.
Make sure that you deal with the negative signs properly
here. Now the second one.
We’ve now reached the difference. Recall that when we first started talking
about the definition of functions we stated that we were only going to deal
with real numbers. In other words, we
only plug in real numbers and we only want real numbers back out as
answers. So, since we would get a
complex number out of this we can’t plug -10 into this function.
[Return to Problems]
(c)
Not much to this one.
Again, don’t forget that this isn’t multiplication! For some reason students like to think of
this one as multiplication and get an answer of zero. Be careful.
[Return to Problems]
(d)
The rest of these evaluations are now going to be a little
different. As this one shows we don’t
need to just have numbers in the parenthesis.
However, evaluation works in exactly the same way. We plug into the x’s on the right side of the equal sign whatever is in the
parenthesis. In this case that means
that we plug in t for all the x’s.
Here is this evaluation.
Note that in this case this is pretty much the same thing
as our original function, except this time we’re using t as a variable.
[Return to Problems]
(e) and
Now, let’s get a
little more complicated, or at least they appear to be more complicated. Things aren’t as bad as they may appear
however. We’ll evaluate first.
This one works exactly the same as the previous part did. All the x’s
on the left will get replaced with . We will have some simplification to do as
well after the substitution.
Be careful with parenthesis in these kinds of
evaluations. It is easy to mess up
with them.
Now, let’s take a look at . With the exception of the x this is identical to and so it works exactly the same way.
Do not get excited about the fact that we reused x’s in the evaluation here. In many places where we will be doing this
in later sections there will be x’s
here and so you will need to get used to seeing that.
[Return to Problems]
(f)
Again, don’t get excited about the x’s in the parenthesis here.
Just evaluate it as if it were a number.
[Return to Problems]
(g)
One more evaluation and this time we’ll use the other
function.
[Return to Problems]
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