Review : Functions
In this section we�re going to make sure that you�re
familiar with functions and function notation. Both will appear in almost every
section in a Calculus class and so you will need to be able to deal with them.
First, what exactly is a function? An equation will be a
function if for any x in the domain of the equation (the domain is all
the x�s that can be plugged into the equation) the equation will yield
exactly one value of y.
This is usually easier to understand with an example.
Example
Determine if each of the following are functions.
(a)
(b)
Solution
(a) This first one is a function. Given an x
there is only one way to square it and then add 1 to the result and so no matter
what value of x you put into the equation there is only one possible
value of y.
(b) The only difference between this equation and
the first is that we moved the exponent off the x and onto the y.
This small change is all that is required, in this case, to change the equation
from a function to something that isn�t a function.
To see that this isn�t a function is fairly simple. Choose
a value of x, say x=3 and plug this into the equation.
Now, there are two possible values of y that we
could use here. We could use
 or
 .
Since there are two possible values of y that we get from a single x
this equation isn�t a function.
Note that this only needs to be the case for a single value
of x to make an equation not be a function. For instance we could have
used x=-1 and in this case we would get a single y (y=0).
However, because of what happens at x=3 this equation will not be a
function.
Next we need to take a quick look at function notation.
Function notation is nothing more than a fancy way of writing the y in a
function that will allow us to simplify notation and some of our work a little.
Let�s take a look at the following function.
Using function notation we can write this as any of the
following.
Recall that this is NOT a letter times x, this is
just a fancy way of writing y.
So, why is this useful? Well let�s take the function above
and let�s get the value of the function at x=-3. Using function notation
we represent the value of the function at x=-3 as f(-3). Function
notation gives us a nice compact way of representing function values.
Now, how do we actually evaluate the function? That�s
really simple. Everywhere we see an x on the right side we will
substitute whatever is in the parenthesis on the left side. For our function
this gives,
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