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Current time:0:00Total duration:6:39

Video transcript

as a little bit of a review we know that if we have some function let's call it f we don't have to call it f but f is the letter most typically used for functions then if I give it an input a valid input if I give it a valid input and I use the variable X for that valid input it is going to map that to an output it is going to map that or produce given this X it's going to produce an output that we would call f of X and we've already talked a little bit about the notion of a domain a domain is the set of all of the inputs over which the function is defined so this is the domain here if this is the domain here and I take a value here and I put that in for X then the function is going to output an f of X if I take something that's outside of the domain within a different color if I take something that is outside of the domain and try to input it into this function the function will say wait wait I'm not defined for that thing that's outside of the domain now another interesting thing about another interesting thing to think about and that's actually what the focus of this video is okay we we know the set of all the valid inputs that's called the domain but what about all the set of all of the outputs that the function could actually produce and we have a name for that that is called the range of the function so the range the range and the most typical there's actually a couple of definitions for range but the most typical definition for range is the set of all possible outputs so you give me you input something from the domain it's going to output something and by definition because we've outputted it from this function that that thing is going to be in the range and if we take the set of all of the things that the function could output that is going to make up the range so this right over here is the set of all possible all possible outputs all possible outputs so let's make that a little bit more concrete with an example so let's say that I have the function f of X defined as so once again I'm going to input X's and I have my function f and I'm going to output f of X and let's say the stuff in the function definition here the thing that tries to figure out okay give it an X what f of X do I produce the definition says f of X is going to be equal to whatever my input is squared well this is a little bit of review we know what the domain here is going to be the domain is a set of all valid inputs so what are the valid inputs here well I can take any real number and input into this and then I take any real number and I can square it there's nothing wrong with that and so the domain is all real numbers all all real all real numbers but what's the range let me I'll do that in a different color just to highlight it what is going to be the range here what is the set of all possible outputs well if you think about it actually to help us think about it let me actually draw a graph here of what this looks like what this looks like so the graph of f of X is equal to x squared is going to look something like this so it's going to look it's going to look something like this I'm obviously hand drawing it so it's not perfect it's going to be a parabola with a with a vertex right here at the origin so this is the graph this is the graph y is equal to f of X this of course is the x-axis this of course is the y-axis so let's think about it what is the set of all possible outputs in this case the set of all possible outputs is the set of all possible y's here well we see Y can take on any non-negative value Y could be 0 Y could be 1 Y could be PI Y could be e but Y cannot be negative so the range here is the range we could well we could say it a couple of ways we could say f of X let me write it this way f of X is a member of the real numbers such that is such that f of X is greater than or equal to zero we could write it that way if we wanted to write it in less Matthieu notation we could say that f of X f of X is going to be greater then or equal to zero f of X is not going to be negative so any non-negative number that the set of all non-negative numbers that is our range let's do another example of this just to make it a little bit just to make it a little bit a little bit clearer let's say that I had let's say that I had G of X let's say I have G of X I'll do this in white let's say it's equal to x squared over X so we could try to simplify G of X a little bit we could say look if I have x squared and divided by X that's going to that's the same thing as G of X being equal to X x squared over X is X but we have to be careful because right over here we have two in our domain X cannot be equal to zero if X is equal to 0 we get 0 over 0 we get indeterminate form so in order for this function to be the exact same function we have to put that because it's not obvious now from the from the from the definition we have to say X cannot be equal to 0 so G of X is equal to X for any X as long as X is not equal to 0 these two function definitions are equivalent and we could even graph it we could graph it it's going to look I'm going to do a quick and dirty version of this graph it's going to look something like this it's going to be it's going to have a slope of 1 it's going to have a hole right at 0 because it's not defined at 0 so it's going to look like this so the domain here the domain of G is going to be X is a member of the real numbers such that X does not equal 0 and the range is actually going to be the same thing the range here is going to be we could say f of X is a member of the real numbers such that f of X does not equal 0 f of X does not equal 0 so the domain is all real numbers except for 0 the range is all real numbers except for 0 so the big takeaway here is the range is all the positive set of all possible outputs of your function the domain is a set of all valid inputs into your function