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Problem 10 Easy Difficulty

Sketch the graph of an example of a function $ f $ that satisfies all of the given conditions.

$ \displaystyle \lim_{x \to 3} f(x) = -\infty $, $ \displaystyle \lim_{x \to \infty} f(x) = 2 $, $ f(0) = 0 $, $ f $ is even

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Video Transcript

Okay, this is a pretty interesting little problem. We want to graph a function F of X that satisfies all four of these conditions. The limit of the function as X approaches tree is negative infinity. So, here's X equal street. That means as X is approaching three to function needs to be going down towards negative infinity. But the same thing has to happen as we approach tree from the positive side. So as X approaches tree from the positive side, we're going to have to go down towards negative infinity. Uh so, so the line X equals tree is going to be a vertical ascent. Okay, so let's go ahead and draw that anything that I'm drawing in red here is not actually part of the function. Two function will draw in blue when we're done. Okay, so this line X equals three is just a vertical ascent toe. As X is approaching three from this side is going to function is going to go down towards negative vinnie as X approaches free from this side is going to go down towards negative next. What else do we have? Uh the limit as X approaches positive infinity of the function to limit of the function as X approaches positive infinity is too. So, as X approaches positive Finney, that means we keep moving to the right in the positive X direction forever. The limit of the function is going to be too. So that means our function is going to be getting closer and closer to a Y coordinate of two. Okay, near this height here. So, as we're moving into positive X direction to function is getting closer and closer to a height of two? Uh F of zero has to be zero. So let's pencil that in right now when X zero to function is zero. And then uh last but not least. Uh F has to be an even function and even function means uh whatever the graph looks like on the positive X axis side, uh It has to look the same on the negative X axis side. So if you think of the function like Y equals X squared, it goes up like this. Uh Then on the negative X axis it goes up like that is the mirror image. This is the reflections of each other um Across this line right here. Um So we'll worry about that last because whatever we make the function look like over here, we'll just mirror it over here. All right, So, we know everything that we draw in. Blue is going to be our actual function. We know to function has to be approaching negative infinity. So it's going to get lower and lower and lower as X is approaching three. So, it's extra approaches tree. The function is going to be going down towards negative infinity. So it's going to look something like this. Getting closer and closer to that vertical aspirin top line and as extra approaches to re from the positive side, it's going to do the same thing. Now, I got to be careful when I draw this curve. Um I can start drawing it but I wanna pay attention to this, remember the limit as X approaches positive infinity. To limited to function will be to so as X approaches positive Finney as we keep moving in a positive X direction, this function has to get close to a value of two. So let's draw another ascent oak line in here At Y. Equals two. Now let's continue with our function. So as X is approaching positive Finney to function approaches to never hits it. Um It's a nascent open to get closer and closer but it never quite hits it. Um So that's everything that we needed to graph on the positive X axis side. I noticed the function is not defined when X equals straight. So now to satisfy the condition that f is even we just take everything that we did here and mirror it here. So let's put in uh negative one, negative two and negative three. And we might as well, since we're trying to make it look like a mirror image of the positive side. Uh let's put in another vertical ascent. Tope at -3. Okay, so now with our curve uh of course f of zero was zero. Now, as we approach three, we went down towards negative infinity. So as we're approaching uh negative three, we're gonna go down towards negative infinity. Uh And then you can see that we um we're approaching negative infinity from the other side of three. So from the other side, negative three, we're still gonna be approaching negative infinity going down like this. But then as excellent towards the positive numbers, uh to function approaches to, so sex is going to continue here in the negative numbers. We're gonna go back up towards to remember we're doing all this because f isn't even function. So whatever you have on the positive X axis side has to be the mirror image of the negative X axis side. So this is going to come up like this, so this is going to come up like this now that we're done, you can see uh that this piece over here is reflected over here. So all four of these conditions have now been met in the graph of this function.