 In this video, we're gonna consider a function whose rule is determined by a graph. And so what one should understand with when it comes to a graphical function, we're gonna take the input values from the x-axis and then the output values are gonna come from the y-axis. Therefore, when you see a point like this one, x, comma, the y-corner is gonna be the evaluation f of x right there. That's gonna be useful to us as we work through these ones right here. Now, the first thing they'll ask about is the domain and range of this function. So the domain here, domain of f, this is gonna be the collection of all x-cornerments that we see actually appearing on the graph. Let's assume that the entire graph is displayed in front of us right here. Now, if we go to the far, far, far right, the biggest x-corner we get is gonna be over here on the right at one, two, three. So x equals three is the biggest value and it's obtained right here on the graph. On the other hand, the smallest x-corner we can see on the graph is gonna be here at one, two, negative three, obtained by this point right here. So, and since there's no breaks or gaps in the graph along the way, we would say something like the domain of this function is from negative three to three right here. Now, a few things I wanna comment here. Like, if first of all, if there was like some open point right here where it was removed, that's a possibility we'd have to modify the domain and say something like, oh, we're gonna go from negative three to negative two, parenthesis union negative two to three. That's a possibility here. Now, let me kinda quickly remind you about interval notation for a moment. If you write something like bracket A comma B bracket, what this means is you're looking for all x-cordants which are greater than or equal to A but less than or equal to B. Having a bracket here means that the endpoint is included. On the other hand, if you had something like parenthesis A comma B parenthesis, this means that x is greater than A and less than B and the endpoints B and A are not included inside of the interval. You of course can mix and match. You could do something like parenthesis A comma B bracket which would mean that x is strictly greater than A but less than or equal to B. You can have this half open, half closed interval that's a possibility. And then you also have this union symbol where you can glue together disjoint intervals. So for example, we could be taking the interval from negative three up to two and negative two and then we go from negative two up to three right here. This is just removing the point negative two and like that sad, sad kid who didn't get picked on the dodgeball team, negative two would feel very sad in that situation but that is an acceptable domain. Now our example doesn't have any gaps or holes in it so we get one continuum of points. We get x from negative three to positive three inclusive here so this is the domain. Range can be a little bit trickier here because when the domain needs to read the graph left to right with the range you gotta read it up and down but the maximum and minimums can be somewhat obscure or hidden if the graph is complicated. Now the biggest point appears to be right here at y equals two and the smallest point you might be tempted to be like oh, this is the smallest point, no, no, no, no, no, no. That's just the right most point. The smallest y coordinates actually right here at y equals negative two and so these end points right here don't have any bearing on the range necessarily. I mean it is true that this is also the minimum value. This is the minimum, this is the maximum. The range is looking what's the range that the minimum maximum go across here and so clearing off my screen real quick we see that the range of our function goes from negative two up to two because again there's no holes or gaps along the y-coordinate here. The y, the y, yeah the y-coordinates. Now like we mentioned before determining evaluation of the graphical function is pretty nice. If you wanna do f of two you just come along the x-axis until you hit x equals two and then you look at the point right here and like we saw a moment ago at sorry when y equals two, when x equals two, y equals negative two. So we see that f of two is negative two and similarly if we wanted to do x equals three that is f of three, we come and look at the graph when x is three, y equals negative one and so that would then be the evaluation there. f of three equals one. Another one let's take a look at f of negative one that means we would come along the graph, the x-axis until we find negative one, look at the point there as we saw earlier negative one when x is negative one, y is equal to two. So we would then record a positive two right there and you'll listen to a few more f of negative two of what do you see happening there when x is negative two that would be right here on the graph so we look at this point right here. When it comes to a graphical representation the points won't always be integer so do your best to estimate them. That's kind of a defect of graphs. You can see the whole picture but sometimes it can be difficult to determine the exact value here. I'm of course choosing convenient values. When x is negative two the y-corner would be one, positive one so that's the function evaluation there. If we were to do the next one f of negative three my graph's kind of messy right now so let me clean this up. If we did negative three that would be at this where we are on the x-axis coming down here we're gonna get negative three common negative two that's our y-corner. The y-corner is the evaluation and then finally when x equals zero we're gonna be actually at the origin so we can see that f of zero would equal zero in that situation. I already had the answer there. Whoops a daisy. And so that's how evaluation works on a graph. You just look on the graph, find the x-corner and then you record the y-corner. That's the function evaluation. That's all there is to evaluating with a graph there. What about solving equations when it comes to a graph? What if we wanna solve the equation f of x equals one? The thing to remember here is that this is the y-coordinate. We're looking for when y equals one. And so that might, an easy way of doing this would be to draw the horizontal line y equals one. We're drawing the line when y equals one and we look for where does this line intersect the graph. This would happen right here at x equals negative two and it would happen right here. Again, we might have to kind of estimate that one a little bit. What does that look like? It's not, I mean, definitely not an integer. I'm gonna say about negative 0.3. That looks like, that's what it looks like to me x equals negative 0.3. This one's definitely negative two. We saw that one earlier. So by intersecting the line we can then solve this equation x equals negative two and negative 0.3. Well, let's try this again with a different value. Let's solve the equation f of x equals three. Where does the function equal three here? And so we would then, in this case, draw a horizontal line at y equals three. That's gonna be up here, right? Y equals three. Now we're gonna see that our function nowhere intersects the line y equals three. They never touch each other. And because they never intersect that means there's no solution to this right here. We get no solution. You can abbreviate it like that or we can actually write out no solution. Geometrically, there's no solution because the function never touches the line y equals three. Now inequalities, how do we do with inequalities here? So inequalities are handled very similar to how we solve equations graphically. We're gonna start off by drawing the horizontal line y equals zero. Now be aware that's just the x-axis in this case. Let me clean up what we already have on the screen here. If we draw just the x-axis, I'm gonna draw it just to emphasize it. We're looking for those values for which the y-coronent is less than or equal to zero, right? So we're trying to figure out where is the y-coronent less than or equal to zero? That would mean we want to be below the x-axis. Now because it's less than or equal to we also want the x-intercepts themselves. So we can see where are we below the x-axis that happens right here. We're gonna be below the x-axis. We're also below the x-axis right here. So seeing from what we have on the picture right here where when is f of x less than or equal to zero? This is gonna be written in interval notation. We go from negative three to the x-intercept. What do we think the x-intercept is right here? It's between negative three and negative two. It seems to be a little bit closer to negative two than it is to three, but it's not quite, it's pretty close to the midpoint. Let's estimate that as around negative 2.4. That's one point where it's less than zero, less than or equal to zero. We get a bracket right there. Next we're gonna take the union. We put a bracket here because we are less than or equal to zero. If this was a strict less than symbol we wouldn't want to have a bracket. We actually would have a parenthesis right there but because it's less than or equal to we include the x-intercept. The other thing is the other x-intercept here is gonna be at zero and so when is it below the x-axis? This will be from zero to the end of the function which is positive three. So we're gonna go from zero to three and so this would be the solution set to that inequality written in interval notation using unions as is appropriate there. And so that shows us how we can find the domain and range of a graphical function, how we can evaluate the function and how we can solve equations and inequalities by looking at the graph. The key thing to remember here is that the function evaluation is the y-coordinate. If you keep that in mind we can solve a lot of these problems using the graph of the function.