So they give us this function here. They want to answer a couple of questions about it. So the first thing they want us to do is to plot why, along with its derivative, and then explain why are derivative or why are function is going to be 1 to 1. So let's go ahead and first figure out what are derivative is going to be. So if we take the derivative of this, we're going to use power rules. So be three X squared, minus six X. Um, so let's go ahead and plot this with original function. So I went ahead and already did this on Decima. So kind of ignore all this. This will come up later. Um, but the red function is our original function, and the blue function is our derivative on those intervals to to negative, but for 2 to 5, so we can see that the red function. If we were to look at it well, that that looks like it passes the horizontal line test. But since we're in calculus, we can actually use the derivative here to help us out. Because notice that are derivative when X is larger than two is always going to be positive, So the function is always going to be increasing after its initial value. So if we know the function is always increasing, since it's positive, that means, well, there's no way we could have two values equal to each other. So what we've been say over here. So why? So we know why Prime is going to be strictly greater than zero on 2 to 5. So then that means why is increasing on 2 to 5, uh, mayor safe hallways increasing all into the five. All right, so now they tell us to find the inverse of this function. And normally what we would do is we would switch the excess with wise so we'd get X is equal to why cubed minus three y squared minus one. But to actually solve this by hand is extremely hard to do. We have to pull out the cubic root function, which is extremely long and tedious. So since they tell us we can use cast software to help us with this, I'm just going to do that. Eso one website that I found that will actually give me the inverse It's this one right here. Um, so you could see the euro periods, e math, help dot net and then kind of all that. But if you just type in tow some search engine, inverse function calculator, math help, this should pop up. You might find something else, but this is the one that I actually found would give me because a lot of other calculators just kind of, like stopped working after a while. Um, so down here they tell us what that inverse it so you could see they didn't actually simplify it completely, because it's, like, two to the three thirds over two, so that could have been to to the one third. But for the most part, I was kind of leave it like this so you could do a little bit of simplification, but this is going to be what are inverse is, and just in case you don't believe me, um, this right here is that function, and so we can see if we were to plot this with y zero x. Let's get rid of the derivative really fast. We could see that these look like it is actually going to be a mere image across. Why is equal to X. So that's going to come up later on. But just to kind of double check that we plug this incorrectly, we can just look at that. Right. So we have that. Now, Um, until again, if you want to just kind of drop this down, you can, uh, so just pause the video, But let's keep on going. So now they want to find the tangent line of our function app. All right. And so to do this, we can use why, minus, why not is equal to m X minus X on point. So, for so first, we need to figure out what is going to be, um, ffx not well, first X dot is 27/10. They tell us so that we need to find Well, what is f of 27/10 or 2.7. So, um, if we were to plug this in, we should get 2.7. Cubed is, like 19.6. Then we do minus three times 2.7 times to put seven. That gives us negative two point 187 Then we minus one. So that's going to give us negative 3187 over 1000. Right now we have that, um So, actually, let's go in and pluck these in. So we know why not is going to be that it's 3187 over 1000, and then this X not here was 27/10. So now we need to find our slope. So what we need to do now is plug 27/10 in tow. Y promise. I'll just do that over here. Um, so the 3 27/10 squared minus six times 27/10. So let's see what we get for us. 2.7 squared times three is 21.87 And that's actually ologists. Write this down. 21.87 and minus six times 2.7. So that's 16.2. So then we can combine those to get 5.67 or weaken right that as 567 over 100. So we've come up here and use that for our slope. Right? So now that we have that, So that was the answer for C they wanted, um, Let's go ahead and find the tangent line for the inverse at that reciprocated point. So they tell us you steer him one for this. So they're one says that f in verse striven tive at some point X is going to be equal to one over the derivative of F in verse at whatever that point is. So if we were to come over here and plug in f of x, not more f of x start wa So I don't wanna play in that big number, but it's just gonna be f of 27/10 that would come up here, race the sex like that in so f 27/10. So these f in verse in the F councils were gonna be left with one over F prime off 27/10. And, well, we found that right here. So we just need to reciprocate this. So this is going to be 100 over 567. So now we can write out the equation and why, minus why not is equal to M minus X done so we can plug in our slope that we found being 100 over 567 uh, ex not so remember, it's flip flop now because our input is f of x dot Which what was our input again? So it's that right there. So let me erase this. It would be positive 3187 over 1000, and then the input for the output or the inverse would be that 27/10. So then this year is going to be our inverse. Oh, the Tangela of the inverse. At that point right now in party, they tell us the plot Everything, along with those two points and a line segment that combines x not ffx. Not so to the reflected point. So this is going to be, um, negative 3187 over 1000 and then 27 10 to 27/10. Negative. 3187 over 1000. Alright, so we wanna find the line segment here. So first we could start by just figuring out what is the line between these two. And then we can restrict it between these two x values here. And then that would give us the line segment So first, let's find our slope. So Slope is going to be so negative. 3187 over 1000 minus 27/10 All over, Uh, 27/10 minus. So I should be plus now 3187 over 1000. So notice that we factor out that negative from the numerator. We will end up with this expression here, so these will cancel out with each other, and we're just gonna be left with negative one. So that tells us our slope is negative one and then we could just use either of these points to plug it. So let's dio So why minus? So I'll use, uh, this point right here. So it would be why lus no y minus 27/10 is equal to. So our soap was negative. One x plus 3187 over 1000. And so this is going to be our line segment. But remember, we need to restrict our X values down. So X is going to be between 27/10 and negative 3187 over 1000, so this here is our line segment. So now we can go ahead and plot all this. And so again, remember, it doesn't matter which one of these points we choose. I'm just choosing that one because I believe that was the one I plotted when I did this. But you'll get the same line regardless. So I was going and plot all of this. So let me go ahead and show all of these. But then, actually, right here, let me. I forgot to put this down. Would be 27 over 10, and then here this waas negative 3187 over 1000, actually, let's get rid of these first. So we already talked about, um the inverse is being equal to each other, so that's good. So we don't have any issues there on then for the tangent lines. So these two tangent lines here notice that if we got rid of those, these are also going to be in verses of each other. We could see how there's symmetric about why is execs just like the inverse is were, um and if we put that line steppin on here, so actually, if we were to make this more to scale. Then we could see how this is actually 45 degrees across. So, actually, let's make our viewing window. Um, negative five. I've the negative five. So here you can see how this actually forms that 90 degree between these two points here so you can see how this is going to be. Ah, the reflection point. Since though the purple and green are perpendicular and that means we did actually reflected across. Why is it attacks? So, yeah, just to kind of get everything on the same page again. Okay, So you can see here this waas everything. And if I zoom a little bit just to get those functions on there so you could mess around with the screen a little bit more, but I feel this is pretty much good enough for what we are doing here.