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4.3 EXERCISES 3. Find the following: Let fl) 2 "Wd g() gL$(2)] fle(2)] 6 flx(-5)1 xLs(sz)] slf(-5)] 5, (Lg(k)] In Exercises 7-14, find flgk)/ and glf6) :4y2(y ...

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4.3 EXERCISES 3. Find the following: Let fl) 2 "Wd g() gL$(2)] fle(2)] 6 flx(-5)1 xLs(sz)] slf(-5)] 5, (Lg(k)] In Exercises 7-14, find flgk)/ and glf6) :4y2(y 33. qk)35. y(2x" (St 37 . r(t) 3t-3x (2x39. yg() = 641. In your own functions. 42. The gener: and8 fl) = -8r + 9 g6) =fl) =gk) =x10. fk) = 2 g6) = 2 - x I. flx) Vx + 2 g() = &x? _ 6 12. fl) = 9r2 Ix; skx) = 2Vx + 2 13. flr) = Vx + 1; g6r) = =!Explain chain ruleConsider the and their de14. flr) = 8 gkx) = V3 - x Write each fun

4.3 EXERCISES 3. Find the following: Let fl) 2 "Wd g() gL$(2)] fle(2)] 6 flx(-5)1 xLs(sz)] slf(-5)] 5, (Lg(k)] In Exercises 7-14, find flgk)/ and glf6) : 4y2(y 33. qk) 35. y (2x" (St 37 . r(t) 3t- 3x (2x 39. y g() = 6 41. In your own functions. 42. The gener: and 8 fl) = -8r + 9 g6) = fl) = gk) =x 10. fk) = 2 g6) = 2 - x I. flx) Vx + 2 g() = &x? _ 6 12. fl) = 9r2 Ix; skx) = 2Vx + 2 13. flr) = Vx + 1; g6r) = =! Explain chain rule Consider the and their de 14. flr) = 8 gkx) = V3 - x Write each function as the composition of two functions. (There may be more than one way to do this ) 15.y = (5 _ x2)*/s 16. y = (3x2 7)23 17. y = -V13 + Tx 18. y = V9 4x 19. y = (x2 + Sx)i? 262 + 5x)23 + 7 20. y = (xln 3)2 + (xi 3) + 5 Find the 43. (a) 44. (a) Find the derivative of each function defined as follows. 21. y = (&x" Sx? + 1)4 22. y = (2r' 9x)5 23. k(x) = -2(1212 + 5)-6 24. f(x) = -7(314 2) 4 25. s(t) 45(3t 8)32 26. s(t) 12(214 + 5)3/2 In Exerc graph of 45. f(x) 46. fl 27. g(t) = -3V7 6t( 5t4 28. f(t) = 8V4t? + 7 30.r(t) 4t( 2t5 + 3)4 fk)



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Determine the following functions:- i. $\quad f_{1}(x)=|x-1|$; ii. $\quad f_{2}(x)=|2 x+1|$; iii. $f_{3}(x)=|\ln x|$; iv. $\quad f_{4}(x)=\left|e^{x}\right|$; v. $f_{5}(x)=\operatorname{sgn}(x+3)$; vi. $f_{6}(x)=\operatorname{sgn}(3 x+2)$; vii. $f_{7}(x)=\operatorname{sgn}(\ln x)$; viii. $f_{8}(x)=\operatorname{sgn}\left(e^{x}\right)$. \{Ans. i. $f_{1}(x)=x-1, \quad x \geq 1$ $=1-x, \quad x<1$ ii. $f_{2}(x)=2 x+1, \quad x \geq-\frac{1}{2}$ $=-2 x-1, \quad x<-\frac{1}{2}$ iii. $f_{3}(x)=\ln x, \quad x \geq 1$ $=-\ln x, \quad 0<x<1$ iv. $f_{4}(x)=e^{x}$ v. $f_{5}(x)=1, \quad x>-3$ $=0, \quad x=-3$ $=-1, \quad x<-3$ vi. $f_{6}(x)=1, \quad x>-\frac{2}{3}$ $=0, \quad x=-\frac{2}{3}$ $=-1, \quad x<-\frac{2}{3}$ vii. $f_{7}(x)=1, \quad x>1$ $=0, \quad x=1$ $=-1, \quad 0<x<1$ viii. $\left.f_{8}(x)=1\right\}$

So they want us to answer a couple of questions about this function that they give us here. So let's just go ahead and start. So they first Selves to plot. Why and why? Prime and explain why. Why is going to be one toe one on this, uh, interval? So first, let's go ahead and find what? Why Prime is so we first just need to take the derivative of this. So first I'm gonna rewrite why? As so why is equal to three x minus to raise to the one half. So if I want to take the derivative of this, I can use power rule. So why Prime is going to be remember, Powerful says, move the power out front. So the one half three x minus two. Then we subtract one from the power. So it's gonna be negative one half and then we have to multiply by the derivative on the inside of this. So the derivative of three X minus two is just going to be three. So we can rewrite this as let's see 3/2, and then that negative one half power just means we send it to the denominator and again. Half power is the square root. So three X minus two. So this is why crime. So let's go ahead and plot both of these. So I went ahead and already plotted. Why? And all we need to do now is locked our other one we have on. Then let's put the interval down. Okay, so this is going to be our interval for this. So they told us to explain why this is going to be 1 to 1. So I mean, you can kind of zoom out a little bit to get all of our blue lines. Let me change. Our balance of us would be zero to Bye. Uh, a little bit ticket. Okay, so that's looks a lot better. So one we could just see from here that the red line is one toe one. But another way we can actually show this is notes that are derivative is always positive on this interval. So since it's always positive, that means our function is always increasing. And if it's always increasing, then, um, that means we have to have a function that is one toe one as long as it's continues. And we could see that our function here is definitely continues over its domain. So ah is going to be one toe one, because why prime strictly greater than zero on two thirds x toe four. So it is always increasing. All right, so now they tell us we want to find the universe of why, and we're gonna call that G. So let me just go ahead and rewrite this down again. So it's why is equal to the square root of three X minus two. So we want to solve for X. So the first thing we could do is square each side. So that's going to give us why squared is equal to three X minus two. Then we can add to over. So we're going to get why Squared was to, um is equal to three X and then we would divide each side by three. So we're going to get X is equal to y squared, plus two over read. And then at this point, we could just change out the wise for the exes and then change out the why actually, for G of X, so it would be G of X is equal to X squared, plus 2/3. So this is going to be our inverse of that function there. All right. Now they want us to find the tangent line of F at X dot f of extra. And so in this case is going to be three f of three. So we're going to plug everything into the line. Why? Minus Why not is equal to m X minus X not eso. We need to find our Why not? And our slopes? Let's go ahead and do that off on this side here. Eso We're just going to take three and plug it into our derivative. So that's going to give us a why Prime, when X is equal to three, is going to be 3/2. So it would be square root of nine minus two, which is going to be three over to route seven. So that's our slope, so we can plug that in right here, 3/2 groups, seven. And then we're gonna have X minus. When we plugged in three. And then why? Minus what? We need to figure out what this is when we plug in three into here. Eso Why is equal to so on X is three. So the square root of nine minus two. So that's just gonna be route 70. So this gives us route seven right here. Okay, so this is our tangent line. Uh, and you could go ahead and solve for why, If you want, but I'm just gonna leave in this form here. Um, then they want us to find the tangent line to our function g at f X, not and all that. So they tell us to use their on one to find the slope of potential light. So this is the room one here. So in this case, though, if we were to do this, we're going to get because remember, if we write this out, I'll do this in blue over here, G is equal to f inverse of X. If f is. Why? Because that's essentially what we solved for over here in part B. All right, so if we want to find this, we're in putting f X. Not so it's going to be f in verse, our area. I'll just plug in G here. So, g um, very or not, it's probably better to just write as F and first just so we don't get things confused. So the f of X not and then this is equal to one over so f prime of so f inverse of be Okay, well, that's really going to be g of our well, first, let's just right is f inverse of So our x not is three. Okay, so, uh, not three. Yeah, of three like that. So this is going to be won over. So f Emerson f just comes out with each other and we have one over f prime of three. And we actually found f prime of three right here. So this was actually f prime of three are very yet. I should probably say right here was a crime of three since we actually simplified it down. So all we need to do is actually plug this in two down here, so it would be one over three over to route seven, And if we were to reciprocate that, that gives us to route 7/3. So now if you want to find this tangent point, So let's see, so f of x not This was, uh What do we get for that again? Route seven And then this is three. So we can also plug this in to the point slope, form her wine, and that would give us So let's just do Why minus why not go to him? X minus X Not. All right. So our why not in this case, was three. Because remember, we flip them because this is our why still, this is our ex still. And then our slope we found to be to route 7/3. And then our X was Route seven. So then this here is going to be that tangent line that we got all right. And so now for the next part, they want us to plot effigy three identity, the two tangent lines and the line segment joining X not f not to f not X, not eso. Let's see what we still need. Toe plot from all So we still need to plot G. Let's do that. Eso What was that again? So X squared. Probably to buy plus to divide by three. But we need to figure out what our new balance for this they're going to be. And we could do that in a second. Um, then we need why is equal to X, and then we also need our tangent lines. Eso that's going to be these two functions here. So this will be why minus three is equal to to route seven over three X minus seven and then down here. Uh, so just copy this. And now instead, we have X over here. Why over here and then this side have three over to route seven. So get rid of this on that side are those or anything else we needed to plot? Um, so f d the identity, the two tension lines and the line segment joining the point. So first, let me get rid of this derivative here and now. So looking at so if we were actually to get rid of all these other one true first and just looked at this green line because you can see that this isn't a 1 to 1 function. So if we were to just kind of restrict this to Win X is strictly larger than zero, then that at least gives us something that is one toe one on, At least for this problem finding the rest of this what we could really do is just look it here so we can see that the output ends at, like, 3.62 and then over here at zero. So we could actually restrict this even further without working too hard. Do 3.6 to where we could really just plug for in tow here. So, actually, let's do that. Eso It would be three times four minus two, and then we take the square root of that. So that's just route 10. So we were to put groups 10 here. So now you can see we have that function. So, actually, let me mess with this real fast to get a better viewing window. Okay, so here is a better viewing window of this and we can see here are actually they want us to pluck these two points. Also, let me do that before I forget. So the points were Route seven three. Then we also had three Route seven. So they wanted us to talk about. So let me just do that. So I want us to talk about any symmetries we see. So first, just between our original lines here are the Red line, which was the original and the green, which is our inverse. We can see how these air symmetric about Why is he could X and these points? If we were to just draw a line segment going straight between them, then it would be like 90 degrees it would form. May perpendicular line with that. Why is it tax like we just went straight across? Uh, now let me get rid of these so we can look at the tangent lines. We can see that the tangent lines are also symmetric about the line. Why is equal to X? So now if we were to put all of this on here like they were saying, we could see how everything is kind of like mirrored across this line. Why is equal to X?

In this problem will be investigating a function, its inverse and both of their derivatives were given the function Y equals the square root of three x minus two, Which is graft on the right over the interval, 2/3-1 4th. Also, this function is considered F of X. A given point On our function is x equals three. Should be right about there will consider that point later. The derivative of this function is also graphed over the same interval and the derivative was found using a combination of the chain role and power. In order to find the derivative, we'll have to rewrite our function, rewriting the square root for that radical as an exponent. We can create an equivalent expression Which would be three X -2 To the 1/2 power. Then using the power rule, we can take the exponents which would then become a coefficient, multiply that By what is in the parentheses three X -2 and subtract one from the exponents, which will become negative one half. Then we multiply it considering the chain rule by the derivative of what is within the parentheses And that derivative of three X -2 would just be three. This derivative can be rewritten with three x minus two in the denominator of a fraction, since it's a negative exponents. And we can rewrite the square root the one half. We can just take that too and put it in the denominator of our fraction and then take the three and put it in the numerator. That is the derivative of our function F of X. Next we'll consider the inverse of our function of X to do so well we can rewrite our function which was why equals The Square Root of three X -2. To find the universe. We must solve for X. We must isolate it to do that. We can square both sides in order to eliminate that square root. So then we have Y squared because three X -2. Then to isolate that term three X. We had to to both sides which gives us y squared Plus two equals 3 x. Then to isolate X. We divide both sides by three, which gives us an inverse of Why squared plus two divided by three. We're going to consider this and call this inverse G of X. Or just G. And in order to graph this and further work with this inverse, we're going to write an X in place of the why it's really G of X. Is X squared plus two divided by three. Next we're going to find the tangent line to our original function F of X at a specified point. So our F of X. That function again was a square root Of, three X -2. We were given an X value of three. We'll find the Y value by substituting in three into our function. So F at three, Subsequute in three for X. And further simplify So we can multiply together those two threes have 9 -2 screwed of 9 -2 and then subtract two from nine and we have this crowd of seven. Really, this is our why value in order to find that tangent line will use the derivative Which we found earlier. So we won't have to solve for that again, really, that was three divided by two times the square root of three X -2. We'll be using that to find the tangent line. Okay, We'll use that derivative and plug in the given X value of three. So putting three in place of X within our derivative of our function. We can further simplify multiplying the threes together to have 9 -2 Within our square root And then subtract two from 9 and our slope of the tangent line, It's three divided by two times the square root of seven. Again, this is going to be the slope of our tangent line in order to create a line a linear equation in the slope intercept form. We'll use the slope formula which slope is equal to the difference in our Y values over the difference in our X values. We know the slope and we know one X. Volume one Y value. So we'll plug in, substitute in everything we know, You know the slope is three Over two times the square root of seven. We know one of the Y values that we recently found was the screw to seven and we know one of the X values is three. We will want to really isolate the Y. In order to put this equation in slope intercept form. So the first thing we want to do is multiply both sides by X -3, right on the left side. Then we'd have three divided by two times a screw to seven X -3 in parentheses. These would cancel out on the right side. So we just have why minus the square root of 7? We can distribute. This really is just a number that's on the outside of our parentheses. So we have three Divided by two times a skirt of seven x. And then -9 with the three times a three in the numerator Divided by two times the square root of seven. Everything will be the same on the right side. Lastly to isolate why and have this in slope intercept form, we'd add the square to seven to both sides, which would eliminate it on the right. And rather than try to combine these two constants, we will just Write it as is so three Divided by two tons of screw to seven x -9, divided by two Times Square root of seven Plus the square to seven equals y. That is our tangent line equation at the specified next. We'll find the tangent line of our inverse function G at a specific point. We use a slightly different method this time we'll use this theorem. One also known as the derivative rule for in verses where this B is coming from within our theorem actually goes back to that original specified point we had for F. Of X. Where the X value is equal to three. Really that can be considered a. And are y. value was the square to seven. Really that can be considered b. So if you have the point A. B. Really? Which what number will be plugging in for B? We had an X. Value A. Of three and R. B. Value or why value of the screwed of seven. So we'll use there in one to find the slope which is really saying the derivative of our inverse assessed at Point B, which will be the square root of seven is equal to one over the derivative of our function F assessed the inverse of function F at B. So let's do that part by piece by piece. Next we find the tangent line of our inverse function, which we called G. Again, G is equal to X squared plus two divided by three. This time we'll use a slightly different method. Well, you steering one which is known as the derivative rule for in verses The M one is assessing the derivative of our inverse at point B, Which is equal to one over Within this parenthesis is the inverse at B. Of our function of and then the derivative of our function F. So where that B is coming from Be really is the Y value that we had that we saw for earlier, which was the square root of seven. That why value can be considered B. And the X value three can be considered a. So the point A. B really is that specified point from earlier 3? The Square Root of seven. So first thing we'll do to find the slope is will assess our inverse of function F. And G is the inverse function Yes. At B. Which is the square root of seven. So the inverse of function of I screwed seven Will substitute seven in in place of X. So the square root of seven squared plus two divided by three. We can simplify that turns into seven Plus two divided by three. Further simplifying, we have three. We have nine divided by three which is the same as three. Okay. That result can go into our theorem really within the outer parentheses. So we have one over the derivative of f assessed at three. That result that we just found and the derivative of the function of at three. We actually already found. That's when we found the slope of the previous tangent line. We assessed the derivative at three, substituting 3 and for X. And we got three divided by two times the square to seven. So we'll write that into our here 1/3 over to Times The Square Root of seven. To simplify this further. Since we have one divided by this number at the bottom really a fraction that will cause the fraction to reciprocate. The denominator will then become the numerator and the numerator will become the denominator the most we can simplify. This is too times the square to 7/3. Really, that will be the slope of the tangent line of our inverse function. G. Again we'll use the slope formula slope is equal to the difference in why values over the difference in X values. And we do know our slope. We just found that as two times a skirt of 7/3. Be careful here because actually are why value it's the inverse the opposite of the previous function. Are why value is 3? And our X value that we know Is the square root of seven. So we have Y -3 Over X- the Square Root of seven. Okay, we want to isolate why. So we multiply both sides by X- the square root of seven. Right? It would be eliminated. Then on the right side You have to times square to 7/3 in parentheses, X minus the square root of seven, Calls Y -3. We can distribute the number on the outside of the parentheses. So we have two times a screw to 7/3. X minus Again. This should be the square root of seven. So because the square to seven is being multiplied by itself. Really we have two times 7/3. Why minus three. I will combine that too, Times seven within that numerator rather than rewriting everything, just become 14 and then we add three to both sides. Hawaii is by itself and our linear equation of this tangent line is in slope intercept form. Right? This time we will at our constants together because we can easily rear 83 As 9/3. That is an equivalent number. So the tangent line is two times the square root of 7, 3/3 X. When we combine these constants, we have minus 5/3 equals why other tangent line equation. Lastly, we'll look at all of our functions and tangent lines graph. So within this graph we have our function F. Of X. As the red curve. We also have G. Fx as this purplish blue curve. Both of the black lines are are tangent lines that we found, Both containing that point either three, The square root of seven Or the squared of 7 3. Another thing that is graft is this identity function Y equals X. Or identity line rather. And what will notice is those tangent lines are reflections of each other over this line. Y equals X.

In this problem will be investigating a function, its inverse and its derivatives. Well, let's not be looking at all of those on a graph. So you'll need to graph this on a c A S, which is basically a graphing calculator. Either hand held or on live. So we're giving this function Y equals the square root of three X minus two. Which we can also consider F of X F of X is graft on the right really at this red curve Over the interval 2/3 to 4. This functions derivative is also graphed as the blue curve over the same interval. And this derivative was found using a combination of the chain role and the power. In order to find this derivative, we really have to consider our function F a bex. And we have to change that square root into an export. So we could rewrite fx as Within Parentheses. three x -2. To the one half. Power Okay then considering the power role, we can take our exponents one half and that becomes a coefficient. Everything that's within the parentheses will stay there three X -2. And we subtract one from the exponents negative 100. Then considering the chain role, we'll want to find the derivative of everything that's within the parentheses and multiply that by what we previously found. So really we're just multiplying it by three. This derivative can be rewritten, the three can become a number within a numerator. The one half means we can put it to in the denominator. And because of this negative one half, really we can change that back to a square root within the denominator. Within the square root three X minus two. That is our derivative. Considering the function that's on the graph. Again, We know that this function f of X is 1-1 really because if we drew a horizontal line, anywhere within within this graph, It would only cross or intersect FFX at one point. This function passes the horizontal line test so it must be 1-1 next. We want to find the inverse of that function really. F of X again, was Y equals the square root Of three X -2. Ultimately, to find that in verse, we will solve for X isolated and that will be our function G. So to start off, we can square both sides to eliminate the square root on the right side. So we have y squared equals three X minus two. To isolate that three X term, we can add to to both sides, eliminating minus two on the right side of me. If y squared plus two equals three X. To isolate the X. We will divide both sides by three. We're left with just X on the right side and we have Y squared plus two divided by three because X. So we're going to consider that in verse. We'll call it G. And what does rewrite G? So that in place of the why there is actually X where X squared plus two divided by three. That is our inverse of Quebec's. All right? So next we'll want to find a tangent line, a tangent line to that function F of X at a specified point. So we were given an X value really, X zero Is equal to three. Remember that F of X was the square root Of three X -2. Okay, we can find our why value using that X value will substitute in the three into our function to then No, both the X and Y value of our specified point. So three is substituted in for X. And we can continually simplify three times three gives us nine nine minus two. Give us the square root of seven. So our point is three Square root of seven. Uh huh. Next we want to find the slope of this line, this tangent line to do that. We'll use the derivative which we already found earlier. We found the derivative to be three over two Times The Square Root of three X minus two. Since we're at the specified 0.3 square to seven, we'll take the X value substituted into our derivative and that will give us the school of our tangible. Okay, so again the slope is three Over two times the square root of three times 3 minus two. And we'll simplify that as much as we can really just what is within the radical three times 3 is nine And then 9 -2 is seven. So really our slope is 3/2 Times A Screw to seven. We want to have this tangent line in the slope intercept form where we have the slope, we have the Y intercept. So to do that, we will use the salt formula to help us create that equation. You'll remember the slope formula really the slope is equal to the difference in why values over the difference in X values. Well, we know the slope and we know it X value and we know why about you? So we can substitute in all those values that we know. You know the slope is three Over two times the square root of seven. We know why value to be the square root of seven and we know it the x value to be three. We'll take this equation and we will solve for y will isolate why? So this is in slope intercept form. We can start off by multiplying both sides by X -3. This will allow us to eliminate it on the right side on the left, we have 3/2 times the square to seven Times X -3. Yes, ultimately we will distribute that on the right side. We have why minus the square of the seven. So let's do that. Let's do that distribution. We have 3/2 times a scroll to seven x minus nine. We multiply three in the numerator, times 3/2 times a screw to seven, that is equal to y minus the square to seven. Lastly to isolate Y Have it by itself on one side will add the square to seven. So both sides of this equation. Now we could add those two constants together. It would be rather complicated to find a common denominator and do so. So we will just leave two constants within our equations. Ultimately, The equation for a tangent line is three Over two times a screw to seven. X minus nine over to tend to scroll to seven Plus the square to seven equals why? This is the tangent line to our function. F A bex. Okay, next, we'll find another tangent line. But this time of the inverse function that we found earlier, we'll also use a slightly different method to find the slope for our tangent line using this hero. So let's start off by remembering that are inverse function G was equal to X squared Plus two divided by three. We could also consider G as the inverse of F. Okay, so if you look at their and one also known as the derivative rule for in verses. Really, that's having us assess our universe function at the the result of that will assess that the derivative of our function of and that's all under one. So this theorem will allow us to find the slope of the tangent line, which in turn we can find the whole slope intercept equation. So considering what B. Is the is if we go back all the way to that point, the specified point from earlier, really can think of the A vow, the A. Or the X. Value as three and are Y. Value as B. So we'll take The square to seven. B is just another way to say the y value. And that's what we will use within this year. So let's start off by assessing our inverse or assessing G. At The Square Root of seven. So we're in verse At the square to seven Will substitute in the square to seven in place of X. So the skirt of seven squared plus two. All over three. We square the square to seven, it just turns into seven Plus 2/3. 7-plus 2 gives us nine And then nine divided by three gives us three. So everything within that parentheses just turns into three using this theory and to find the slope. So really we just have the derivative of our original function F assessed at three. And we actually did that earlier as well. If we go back up two right here, when you're finding the slope of the other tangent line, We assess the derivative at three, we found that result to be three Over two times to screw to seven. So let's put that in place in our theory, 1/3 Over two times the square to seven. So the last thing we have to do here because all of that is in the denominator underneath. The one. Really, what happens is that fraction in the denominator reciprocates the numerator becomes the denominator, the number the numerator and the denominator becomes the numerator and our slope of this line would become too Times Square to 7/3. Maybe that's the slope of the tangent line. We'll use the same method as before using the slope formula to then in turn create a slope intercept equation for that tangent line. So the slope is the difference in our Y values over the difference in our X values. We just found the salt to be too times the square root of 7/3. And be careful here because this is an inverse function, The y value and the original x value switch places. So really the y value that we know will be three and the X value that we know is negative step. Okay, so let's simplify that equation really. Let's solve for why You can start off by multiplying both sides by X- the square root of seven to eliminate it in the denominator on the right side. Okay, so we have two times a screw to seven over three, 10 x minuses square to seven all on the left side of the equation. On the right side, we have y minus three. We will distribute that number that fraction that's on the outside to our terms on the inside We have two tons of skirt of seven over three X. Then we have to Times the square to seven times the square to seven in our denominator really, which turns into two times seven over three equals y minus three. Okay, we can just further simplify that two times seven will give us 14 in the numerator. Finally to isolate why have this in slope intercept form will add three to both sides. Right? These two constants negative 14 3rd and three are much easier to combine considering one is a whole number. The other one is a fraction without any radicals. So let's do that. Let's find the common denominator of three. Really, we can change the number three into nine thirds and then we can combine those two fractions. So are tangent line equation is to times a scroll to 7/3 X. Really, when those are combined we have minus five thirds. That is equal to why that is the tangent line at our specified point of our inverse functions. G. Alright, last thing we will do is look at all of those functions. Tangent lines, we will look at them on a graph. So you can go ahead and graph those on your C. A. S. Or graphing calculator. We'll start off with this red function. This curve is F. Of X. The purplish blue curve is G. Or the inverse. Oh yes, we have both of the tangent lines that were found. Those are these black lines, Black linear lines, both going through those specified points. One of those points being square 273 The other one being 3 the square root of seven. Another thing graft is our identity. Really this linear equation Y equals X. And when we look at the tangent lines graft really, we can notice that they are reflections of each other. They reflected over that identity line Y equals X.

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.


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