Introduction to the domain and range of a function
- 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, that 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 product an output that we would call "f(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 if this 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(x). If I take something that's outside of the domain, let me do that in 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, "hey, wait wait," "I'm not defined for that thing" "that's outside of the domain." Now another interesting thing to think about, and that's actually what the focus of this video is, okay, we know the set of all of 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 have outputted it from this function, 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(x) defined as, so once again, I'm gonna input x's, and I have my function f, and I'm gonna output f(x). And let's say this def... The function definition here, the thing that tries to figure out, "okay, given an x, what f(x) do I produce?", the definition says "f(x) is going to be equal" "to whatever my input is, squared." Well, just as a little bit of review, we know what the domain here is going to be. The domain is the set of all valid inputs. So what are the valid inputs here? Well, I could take any real number and input into this, and I could 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? Maybe 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, actually, to help us think about, let me actually draw a graph here. Of what this looks like. What this looks like. So the graph of "f(x) is equal to x squared" is going to look something like this. So, it's gonna look, it's going to look something like this. I'm obviously hand-drawing it, so it's not perfect. It's gonna 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(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? Well 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 zero, y could one, 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(x)", let me write it this way. "f(x) is a member of the real numbers" "such that, is such that f(x) is greater than" "or equal to zero." We could write it that way, if we wanted to write it in a less mathy notation, we could say that "f(x) is going to be" "greater than or equal to zero." f(x) is not going to be negative, so any non-negative number, 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(x), let's say I have g(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(x) a little bit, we could say, "look, if I have x squared" "and I divide it by x, that's gonna," "that's the same thing as g(x) being equal to x." "x squared over x" is x, but we have to be careful. Because right over here, we have to, in our domain, x cannot be equal to zero. If x is equal to zero, we get zero over zero, we get indeterminate form. So in order for this function to be the exact same function, we have to put that, 'cause it's not obvious now from the definition, we have to say, "x cannot be equal to zero." So g(x) is equal to x for any x as long as x is not equal to zero. Now these two function definitions are equivalent. And we could even graph it. We could graph it, it's going to look, I'm gonna do a quick and dirty version of this graph. It's gonna look something like, this. It's gonna have a slope of one, but it's gonna have a hole right at zero, 'cause it's not defined at zero. So it's gonna 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 zero," and the range is actually going to be the same thing. The range here is going to be, we could say "f(x) is a member of the real numbers" "such that f(x) does not equal zero." "f(x) does not equal zero." So the domain is all real numbers except for zero, the range is all real numbers except for zero. So the big takeaway here is the range is all the pos... The set of all possible outputs of your function. The domain is the set of all valid inputs into your function.