suppose-f-1-is-the-inverse-function-of-a-differentiable-function-f-and-let-gx1-f-1x-if-f32-and-fprim

Question

Suppose $f^{-1}$ is the inverse function of a differentiable
function $f$ and let $G(x)=1 / f^{-1}(x)$ . If $f(3)=2$ and
$f^{\prime}(3)=\frac{1}{y}$ , find $G(2)$ .

Amy Jiang · Accepted Answer

Okay, so let&#x27;s start by finding for us to find you prime at two. So that&#x27;s ever finding, um, are derivative of G. So we can use our Kocian role. That&#x27;s preservative of one, which is they&#x27;re all and then minus distributive of this time at X and in times one over f inverse, that&#x27;s X quiet. Okay, so we know that f evaluate at areas you could too. So since our function is, um, it has an inverse function. We also know that given this we have a inverse at two. What s equal to three. Well, that&#x27;s what we need to find you prime at two. What? I see four to negative F in verse. Derivative at you over f been Ruth at two. Glad. Okay. So the derivative of our inverse function, we know that Well, that is equal to Hi there. I&#x27;m seven. We have what, over half prime at F in verse at two. This is all over a crime, or it&#x27;s our f in verse right to quit. So let me simplify it out further. So we get negative one over f in this at two when an f crime and converse at two Okay, so we know what? Out of interest. Too difficult to three. So we get three squared in an F crime at three. So f prime at three deaths. With you one overnight, we get negative one over and nine times one overnight, and that&#x27;s this equal to negative one.

Amy Jiang · Answer

Okay. So as to find the following we were called out from their seven we have that&#x27;s equal to the derivative of our function and then are in versed. ATS are points five. Okay, so we know we&#x27;re giving that four is equal to five. So that means inverse of five is equal to four. That gives us one over that crime at four. How did I notice? Well, I know that our function is, um it has an inverse function, so that means our domain of our function is equal. Could range of our in verse on their veins of our function is equal to domain of our Okay. I know f prime at four. That people through to over three. So we have one over to over three and that she put trio

Theodore Zachary · Answer

in this problem were asked to find the derivative of an inverse function at the origin at zero were given that the original function goes through the origin so g of zero equals zero. And we&#x27;re also given that the derivative of the original function at the origin at zero equals two. We don&#x27;t know the specific all phone specific functional format of G, so we can&#x27;t calculate derivatives and use them directly. But we can use the derivative rule of in vs to solve this nonetheless. Remember, the derivative ruled in verses from your book says that the derivative oven in verse evaluated at some value be is the reciprocal of the derivative of the original function evaluated at the value that be maps back to now. Remember, this concept of reversing the mapping is critical, right? If our function G maps the value of zero at the origin over to the value zero, then that would imply that the inverse of G maps This is zero back to the original zero. So in order to use the serum, we have to make sure first of all, the two preconditions are satisfied. You always need to double check that before you apply it here. Um, right requirements are first of all, that the function g have an inverse in the books as it does take that as a given. And it&#x27;s also the other precondition is that the derivative the original function cannot be zero Over the interval of that sin question the interval you&#x27;re working with now here we don&#x27;t have interval other than a single point and we are told that the derivative of G at zero equals two and therefore that second precondition is met so we can use this. So let&#x27;s re specify the general form using a specific number in the problem. So we&#x27;re being asked to find the derivative of the inverse of G at zero and by the derivative rule of in vs that&#x27;s going to equal the derivative of the original function G evaluated at the value that zero maps back to Well, we just showed that the value that zero maps back to is itself zero. So this is going to be equal to the derivative of the original function G at zero. And we were also given that g prime of zero equals two. So this equals 1/2 and that&#x27;s the derivative of the inverse at the origin. It is always useful if you can to use some sort of intuitive a graphical approach to visualize and double check your work. Make sure that, uh, it all hangs together. In this case, we take advantage of the fact that the graph of an inverse is simply the mirror image of the graph of the original function over the 45 degree diagonal. So that would mean that if we have a function se G of X that goes through the origin and as a slope of two at that point than the graph of G in verse is going to be the mirror image of G over the 45 degree line. It will also go through the origin and its slope at the origin. It will be 1/2 and in fact, graphically, that corroborates what we came up with through the general result

Karl Schaefer · Answer

and this problem. We&#x27;re told that the differential function why equals g of ex has an inverse and that the graph of G passes through the origin with slope, too. Where then asked to find the slope of the graph of G in verse at the origin. Someone&#x27;s going to sketch a very quick picture of the situation. So I&#x27;m gonna graph she of X and blue and we really don&#x27;t know anything about it other than at the origin it passes through the origin and the slope of the or at the origin is too. We don&#x27;t know really what it looks like, you know, maybe it comes and looks like this to make the soap equal to two. Maybe it looks like this to make the slope equal to two. We don&#x27;t really know what it looks like, but we know that at the origin the slope is too. So this is just a snippet of Gaea Vex. Let&#x27;s try and decompose What? The problem gives us algebraic Lee. So one thing that the problem tells us is that G of ex passes through the origin. So this tells us like in this graph that g of zero is equal to zero. So if we&#x27;re talking about inverse functions, we can also rewrite this as G inverse of zero is equal to zero. Although this is ah, these air, the zero should be You should think of his switching places here. So I&#x27;m gonna write these all in in a different color. Just tow, make it a little bit more clear. So g of zero is equal to zero means that GI inverse of green zero is equal to blue zero. That&#x27;s how in verses work you you switch the switch, the inputs and outputs. But this tells us that GM zero is equal to zero and were also given that the slope is too at the origin. So that tells us that G prime of zero is equal to two. Now we&#x27;re asked to find this slope of G in verse at the origin. So let&#x27;s turn this into a more calculus looking, you know, symbol, because that&#x27;s going to tell us how to evaluate it. So slope that something about the derivative right, we want to find the slope of G inverse so that&#x27;s related to the derivative of G in verse. But we want to know at the origin. So that&#x27;s at X equals zero. So this is how we translate the slope of the graph of GM versus the origin into amore calculus looking language. So this right here, this is what we want to find. If we want to find this, we don&#x27;t know anything other than anything really about g inverse other than what we know about GI. So we need to use the serum from the textbook from this section which said that if I want to find the derivative of an inverse function like this, then that&#x27;s just the the reciprocal of the derivative of G evaluated at the inverse of zero. So hear this zero is is coming to zero right here is this is fear on one from the section. So now we need to know what G University zero is. But we already know what the universe is. Zero is. We know that G universes zero is equal to zero. So what we&#x27;re looking for is one over the derivative of G at X equals zero. But this is just another way of writing one over g prime of zero. Because that&#x27;s what the derivative of Gs zero is and again in the problem. We&#x27;re told that the derivative of zero assuming the truth of G at zero is too. So we write this in and we get our final answer is 1/2. So to summarize, this derivative that we were looking for to find the slope of GM versus the origin is equal to 1/2.

Linh Vu · Answer

in this question here were given the function. Why, Coach Tree angst. And then the GX has, uh, has the inverse Julian Rosenberg. Thanks. And now on the graph, we&#x27;re busting through the origin so arching. 00 is on. Ah, graph of she ANC&#x27;s. And now we have Does love on this bone will be is an accident. Why here? And I&#x27;m sort of slop and upon a zozo here echo Jew do. And we know that for the slump and has equalled You&#x27;d, uh, the river, the other function And then finally on the ANC&#x27;s If I were in pretty that you bring under zero And the question asked her to find Islam by the slaw. Ah, the graph off you, Bram Jian vers I have the origin so we know this love hand We&#x27;re excited a g in verse, Bram there So like I was right. It&#x27;s no women and d up d thanks for me. Is he an end of only on, uh, ex Echo ju zero dessert here when we deserve for the one hand? Because that&#x27;s inverse because they&#x27;re ICO. So this will be a little bit confusing here and by the formula found on the riveting. I&#x27;m, uh, inverse function, which you get Nichols one off the cheap. Bram. I&#x27;m the zero. And then the tree from Missouri could you to defend. We get equal to 1/2, and that&#x27;s gonna be the slum that we&#x27;re looking for here.

Doruk Isik · Answer

in this problem, there is a function, and we know that that function has a story in that one Warner function processor in the world, given day, everything on your the diameter att, that exact point. So we don&#x27;t have a function passes through this point for when exes, too sanctions those or also meet our narrative emulates that function. Uh, s a group one over three. And we arrested Flying the damaging off dysfunction humors humors X it For now, we know that David all functions worse. It will let that one point in this case that boy is for we&#x27;ll be equal war over functions, narrative able like another. Since we know that dysfunction passes through this point emotional to past that point for on this will be equal to a bunch of their with their privilege. Uh, this point, since this is given right here, and charitable be won overwhelming feet and derivative off its fevers. Evelyn, that, uh, this point will be equal to three

Problem

(a) How is the logarithmic function $y=\log _{a} …

Problem 40 Hard Difficulty

Suppose $f^{-1}$ is the inverse function of a differentiable
function $f$ and let $G(x)=1 / f^{-1}(x)$ . If $f(3)=2$ and
$f^{\prime}(3)=\frac{1}{y}$ , find $G(2)$ .

Answer

$-1$

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Essential Calculus Early Transcendentals

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Catherine R.

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James StewartEssential Calculus Early Transcendentals

Grace H.

Catherine R.

Kayleah T.

Caleb E.

James Stewart

Essential Calculus Early Transcendentals