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