 In this video, we're gonna consider a function that is described numerically. That is, we actually have a table of values that determine the relationship for our function. So our function will be called f and it's gonna have five or six points in its domain. So this function will be defined for x equals 10, 14, 18, 22, 26 and 30. And those are the only values for which this function is defined. So with the first question, what's the domain of the function, we actually can list what the domain is. We can just list all the values. This function's only defined for 10, for 14, for 18, 22, 26 and 30. Those are the only values in its domain. There's only six numbers in the domain of this function. The range of the relationship that the function gives us is that it's gonna make the connection that when x is 10, y, the y coordinate, it's gonna be negative 12. In other words, looking at the first column of the table here tells us that the function evaluated at 10, f of 10 will just be negative 12 because that's what the table tells me. Or if I wanted to do f of 22, I would look at this column right here. When x is 22, we'll assign that to the number one. So f of 22 is equal to one. And that's how we can do evaluation with the table here. So when it comes to the range, right, the range is all the possible output of the function. Or we could think of it as these are all the y's that come associated to x's. We are just, to find the range, we just have to look at the bottom row of this table here. Now we could put them in order if we wanted to, which it is actually in ascending order, which is fine. But the order when you list it doesn't matter too much here. Negative 12, negative six, negative two, one, three and eight. And so this is a very simple way of describing a function relationship. The domain is just the numbers on top. The range is just the numbers on the bottom. If we do function evaluation, we just look up the number on the table. So look at this one right here, f of five. What is f of five? Well, if we look along the table, it's like 10, 14, 18, 22, 26, 30. There's no five, right? And because there's no five in the table, and that's the entire function, then it turns out that f of five is actually undefined. It's not inside the domain. Five's not in the domain. So there is no number associated to five. It just wasn't defined for this function. So we would say that it's undefined. Sometimes we use the acronym D in E, which is short for does not exist. F of five does not exist because for whatever reason, we never decided what f of five should be. And there are a lot of practical reasons why one would not define this function for five. We don't necessarily have to worry about the context of this thing. We are just interested in how is it defined? It's the fives not defined here. So there is no f of five. That's how we're gonna do evaluations on these things. It turns out solving equations with this tabular approach to functions is pretty nice as well. It's pretty simple. So if we wanna solve the equation f of x equals negative six, we have to look for any x-coordinates whose y-coordinate is negative six. So we're gonna search along the bottom row of the function because that's where the f of x's are, that's the y-coordinates. We can see that there's a negative six right here. And so this would tell us that when x equals 14, f of 14 would equal negative six. So that's a solution there. Now you'd have to go look for the whole bottom right because there could be more than one solution maybe. Not in this situation. Turns out there's just the one. And so solving the equation f of x equals negative six means x equals 14. On the other hand, what about solving the equation f of x equals zero? So what this means is we're looking for an x-value such that the y-value is equal to zero. Now as we search the bottom here, we don't find any y's equals zeroes, right? It turns out that for this function, at no time, can f of x equals zero. And so since there is no choice of x that makes f of x equal to zero, that would mean there's no solution. There's no solution to this equation. Sometimes you'll see people draw the circle with a line through it. That means that's shorthand for no solution. Now on the other hand, if we wanna consider the inequality, when is f of x greater than zero, f of x here is the y-coordinates, the bottom row of this table. We wanna look for all values that will make f of x greater than or equal to zero. Well, that's not negative 12, that's not negative six, that's not negative one. Oh, it is greater than zero for one, three, and eight. So we're gonna report the corresponding x-coordinates there. So our solution would be x is equal to 22, 26, or 30. Those values of x will make f of x be greater than equal to zero there. And so that's all one has to do when you solve equations or inequalities with a table function. Or if you just wanna do function evaluations, you just look up the information on the table and you just have to record it. No equations have to be solved because this function is described in a numerical fashion. It's just a table. And so you can see some huge benefits of having this table. Solving equations is easy when you have a table. It just comes down to how difficult it is to search the table. And if the table's not too big, it's not too hard to search. But I should also mention though that if you have a lot of values, you know, tens of thousands of values, it might be very difficult to search the table. And so although it's nice, this numerical approach is very nice, it does have limitations. The bigger and bigger the function gets, the harder and harder this process gets. But I did wanna show you an example of solving equations and evaluating functions when it's in this numerical slash tabular form.