We could graph it, it's going to look, I'm gonna do a quick andĭirty version of this graph. Now these two functionĭefinitions are equivalent. '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. To be the exact same function, we have to put that, We get zero over zero, we get indeterminate form. Because right over here, we have to, in our domain, x cannot be equal to zero. G(x) being equal to x." "x squared over x" is x,īut we have to be careful. 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 To "x squared over x." So we could try to I'll do this in white, let's say it's equal Let's say that I had g(x), let's say I have g(x), Just to make it a little bit, a little bit clearer. Let's do another example of this, just to make it a little bit, 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. "f(x) is a member of the real numbers" "such that, is such thatį(x) is greater than" "or equal to zero." We could write it that Say it a couple of ways, we could say, "f(x)", Y could be pi, y could be e, but y cannot be negative. Of all possible outputs is the set of all possible y's here. So let's think about it, what is the set of all possible outputs? Well in this case, the set Of course is the x-axis, this of course is the y-axis. So this is the graph, this is the graph, "y is equal to f(x)," this It's gonna be a parabola with a, with a vertex right here at the origin. Is equal to x squared" is going to look something like this. 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 But what's the range? Maybe I'll do that in a different color just to highlight it. Nothing wrong with that, and so the domain is all real numbers. So what are the valid inputs here? Well, I could take any real number and input into this, and IĬould take any real number and I can square it, there's The domain is the set of all valid inputs. Is going to be equal" "to whatever my input is, squared." Well, just as a little bit of review, we know what the domain X, what f(x) do I produce?", the definition says "f(x) Here, the thing that tries to figure out, "okay, given an I'm gonna input x's, and I have my function f,Īnd I'm gonna output f(x). So let's say that I have the function f(x) defined as, so once again, Little bit more concrete, with an example. The set of all possible, all possible outputs. Of all of the things that the function could output, that is going to make up the range. It from this function, that thing is going to be in the range, and if we take the set Something, and by definition, because we have outputted Something from the domain, it's going to output Of definitions for range, but the most typical definition for range is "the set of all possible outputs." So you give me, you input The range, and the most typical, there's actually a couple That is called the range of the function. Outputs that the function could actually produce? And we have a name for that. The focus of this video is, okay, we know the set ofĪll of the valid inputs, that's called the domain,īut what about all, the set of all of the Thing to think about, and that's actually what 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 Outside of the domain, let me do that in a different color. If this is the domain here, and I take a value here,Īnd I put that in for x, then the function is A domain is the set of all of the inputs over which the function is defined. To product an output that we would call "f(x)." And we've already talked a little bit about the notion of a domain. It is going to map that, or produce, given this x, it's going "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,Īnd I use the variable "x" for that valid input, it is A review, we know that if we have some function,
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