|I would like to continue on the idea what a function is by pointing out the importance
of the domain and range. Often students think the domain is just
something that we math professors use to torture you, but it is actually very important in many
cases. Many of the functions deal with a domain of all possible numbers, or almost all possible
numbers, or something like that. There is an example that you are quite familiar with.
|Let's define a function: x is the set of people and let's try to say that parent(x) is any (biological) parent of x. This is either the father or the mother. This is not a function. It is
|not a function because everybody has two biological parents and we this as a function.
We would have me trying to map to both my mom and my dad, and this is
|not a function because when I put in x then either I get both my mom and dad out (in which
case would be parents-- plural) or sometimes my mom is the results or
sometimes my dad is the result.
This is no good. But if we try to find a function called for example: Mother of x ( biological mother, I don't want to get in the idea of worrying about adoption or similar things)
|This is the female parent and this is a perfectly well defined function
because everybody has a biological mother, so it is defined for every human person.
You can argue that Adam and Eve were a little bit off on those, but other than that everybody
|has a biological mother and this is a well defined function.
Maybe there are people who don't know who their mothers are, but certainly have mothers, so we are okay there. We can define another function called father the same way and this is a perfectly good function. So, the domain is the set of all people, we can also write that
|x is in the set of people (not x is people), and the range or the collection of all possible outputs
let's say of the mother function its all female people who have had more than one child.
Notice that again both me -- I will go to my mom, and my sister also has the same parent,
|the same mother. That's okay, this is perfectly good function.
(If I try to go from mothers to children, again I have a problem because I would have to say oldest child, youngest child, something like that This may seem a little bit irrelevant to the idea of studying calculus
|or precalculus or math of any sort, but we have exactly the same kind of thing in the case of
for example, the square root where we want to pair our function, so our square root.
Let's write it, call the function "SQR" of x is a number (I need to leave a little space) y so
|that y squared (Y^2) is equal to x. Now, this may seem okay to you they way that I said it because, for example if we try again and figure out what SQR(4) is, well we want
|to find some number so that when we square it I get 4, since 2 squared (2^2) is 4, but just like the idea with mother and father there is another number that would also play
|the game because negative 2 times negative 2 is positive 4, so this is no good, because
I don't know whether my answer should be 2 or minus 2. Often when we solve an equation,
we do something like this, this is no good. This is not really a function, this is a problem.
|We often write plus or minus, so if we are trying to solve: x squared equals 4, then
we write x is plus or minus 2, but plus or minus 2 is not a number, it is two numbers:
it is plus 2 and it's minus 2. So, this is not a square root, so this guy is not a function.
If we want to define this to be a function, then we have to say it is a non-negative number
|y, so that y squared equals this, then SQR of 4 would be 2, so 2 squared is 4 and 2 is not negative, and now this is okay since square root of 4 can be negative 2 because
|negative 2 squared is 4, but negative 2 is a negative number so it is no good, so we are okay. So,
it is important often to say what our possible range of values is or sometimes our domain as well.
|So, it is important to pay attention to both the domain and the range or trying and get some kind of functions. Now, here we are trying to define the outputs and we are saying we want to choose the positive one. We could just as well, let's call this "n square" and then we want to choose a negative number so that y squared is x. And so now n square
|(NSQR) of 4 is negative 2 and this is okay too because when I take the square root I
just want to choose the negative answers. Now, I have two functions that come out with
this idea of square. So, here is an example where choosing the range (or the codomain) matters.
|We are trying to define our function and be careful about this kinds of things sometimes, this will come up quite often in both calculus and pre calculus. In math courses it is important when we are trying to define our function. Let me make a comment about the range, so we said that when we define function, a function pairs our inputs (that are called domain)
|from x in our domain with some y in our range (I'm fuzzy about if I want to use the word "range"). Unfortunately, the word "range" has two different possible uses.
Most people use it to mean, so let's say, I am going to use a different word, I am going to use the word codomain. The codomain is the outputs, the possible outputs
|and then the word image, this is the outputs in which occur. And it is often easy to identify
|the codomain because we just set it up. For example, here where we said it is a negative
number y so that is what I choose for codomain, it is not always easy to identify the image.
Now the word range, some people use "range" in fact many textbooks will use this, the word
|range for the image, the ones that actually occur. But some people would use range and so for example when I was in school we often use this word for the range and so the word range is often confusing because some people would use it to mean all possible set of outputs and some people would use it to mean those that exactly occur. We usually know the codomain,
|or the possible set of outputs up front, but it takes work to figure out what the image is.