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In Exercises $29-38,$ for the function determined by the given equation (a) determine its domain, (b) sketch its graph, (c) determine its range, (d) show the function is one-to-one, (e) sketch the graph of its inverse function, (f) determine the domain of the inverse function, (g) determine the range of the inverse function, (h) find the equation of the inverse function and (i) verify that $f^{-1}(f(x))=x$ and $f\left(f^{-1}(x)\right)=x$.$$f(x)=2 \sqrt[3]{x}$$

(a) $(-\infty, \infty)$(c) $(-\infty, \infty)$(f) $(-\infty, \infty)$$(g)(-\infty, \infty)$(h) $f^{-1}(x)=\frac{x^{3}}{8}$

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 1

Inverse Functions

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11:51

In Exercises $29-38,$ for …

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In Problems $29-38,$ the f…

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In Exercises $31-36,$ (a) …

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In Exercises $23-30,$ (a) …

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So if we want to find the domain of this, well, we know for all the odd routes, there's no bad numbers to plug in. So the domain of this is just gonna be negative. Infinity to infinity. Um, And then to get the graph of it, we can just plug in a couple of values for this, Uh, since we're just essentially stretching the Cuba by two. Uh, so let's go ahead and plug those in. So, first, uh, if we plug in zero, that would just give us zero, and then we have to think, Well, what other numbers could we plug in that have nice cube roots? Um, so, like, negative one would work. Um, so negative one. So that would just give us negative two. And then one also works, which would give us, uh two. And now we can also get some points of our inverse because, remember, if we consider have to have the point XY, the inverse is going to have the point y x so we can just switch these around so x f inverse of X. So this is going to have the point negative to negative 100 and then to one. So let's go ahead and graph this. And so we really only need to go out to I'll just go out one more in all directions. So our function f is going to be negative. One negative two 00 and then one two. And this should look like that. Uh, and I just kind of know that this is what it's supposed to look like because, you know, the cube root doesn't go up like the cute does. Um, and if we would have put some more points, it would have been a little bit more obvious past. So this is going to be why is he going to FX now for the inverse? So this is going to be negative too negative. One 00 And then we have 21 So this one is going to look more like this. And so then this is going to be our inverse. So we have both of those graph now, uh, and I guess we could actually go ahead and do that right now to find the range of F. Well, that's just the green line. Um, so I mean, this just keeps on going forever so we know that also goes from negative infinity to infinity. Uh, then to show this is 1 to 1. I mean, we could just look at the green line and say, Okay, well, this passes the horizontal I test, or we can be a little bit more exact. And actually look at the derivative of this. So let's go ahead and do that. So f prime of X. Well, this is actually supposed to be X to the one third power, so two times X to the one third. So power rule says it's going to be two thirds and then acts to the negative two thirds power. And then, if we were to kind of rewrite this, this is going to be two thirds times one over X squared, cubed rooted, which, Well, if we have X squared, that's always positive. Cuba of that. Always positive. So knows how this is always going to be strictly larger than zero. Um, and even where this is undefined at zero, it would still be positive. Um, in the sense of it would be going towards positive infinity. So because of that, this implies always increasing and functions that are always increasing our 1 to 1. Okay, so we have that now for the domain and range of our inverse. Well, we could look at the graph up here and say Okay, Well, um, that's just gonna be negative. Infinity to infinity for both. Or we can use the fact that we know the domain and range are just flipped from our original function. So the domain of F inverse is the range of F. So that's going to be negative. Infinity to infinity. And then the range of F and verse is supposed to be the domain of F, which is also negative. Infinity to infinity. Uh huh. Next, they want us to find the inverse of this. So we just come down here. We first replaced this with why, uh then we interchange them. So be X is equal to two cubed root of y. And now we just saw for why, so we would divide by two the B X over two. There's even two cubed root of why, and then we would cube each side and that would give us, uh X cubed over eight is equal to y, which is going to be f inverse of X So this is our universe. Now to show that these compositions both give us X, Um, let's just go ahead and start with this first one. So f inverse of f Quebec's is going to be. So we take affects and plug it into FM verse. So it'll be two cubed root of X, cubed over eight. So that would just be eight over X, divided by eight, which just gives us X so that one checks out and then we would do something similar for the other one. But just where it's flip so f inverse of f of X is going to be, uh so f inverse is supposed to be that cubed and then we just plug f of X in, which is two q brew of X, so that would give us eight. Uh, and then we end up with Oh, I just wrote the same thing twice. I was wondering why that look like what we just did. Um, yeah, we want f of f inverse of x. Okay, now, this should actually give us something different. So, um, affects is going to be to Cuba through FX, and then we just plug in F in verse in here, which is going to be X cubed over eight. So that would be two times X over to the two's. Cancel our two special effects so that one also checks out.

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