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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 $f\left(f^{-1}(x)\right)=x$.$$f(x)=(x-1)^{3}$$

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

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 1

Inverse Functions

McMaster University

University of Michigan - Ann Arbor

Lectures

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In Exercises $29-38,$ for …

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In Exercises $31-36,$ (a) …

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In Problems $29-38,$ the f…

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In Exercises $23-30,$ (a) …

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So if we want to find the domain of this function here, well, Xmas one cute, that's just a polynomial. So we know polynomial is always are the domain of negative infinity to infinity. Um, two. And since this is an odd polynomial, we know that this would also go from negative infinity to infinity so we actually can go ahead and get the range just by kind of using some information we already know. Promise. Now, in order for us to graph this, what I'm going to do since this is really like we took the function x cubed and then shifted it one to the right. So essentially, we're going to take the X values and add one to them. Um, if we first figure out how to graph X cubed, then we can just add one to all the X values. So that's what I'm going to do first. So we're going to have x x cubed. And so the nice values are plug into here is gonna be negative. 10 and one. So that would give us negative 101 Uh, and then if we go ahead and shift these over, so that'd be X and then X minus one cube. So the output stay the same, but the inputs we shift over one. So we just had one to them would be 01 and two and something else we can do while we're still here is keep in mind that if these are the points X Y, the inverse is going to have the points Y x So we just need to flip these around to get our inverse. So X f inverse of X is going to be negative. 1001 and 12 So now let's go ahead and graph these, um so X needs to go. I'll just do two in all directions and then I need to go two up and two down as well. So first f is going to be zero negative. One one zero and then to one. And so we know Cubes. Look, something kind of like this and then for our inverse. So this is going to be negative. 100 one and then one two. And this would look something kind of like this here, so I should probably right. This is why is it a to f of X And then this is why is the U two f inverse of X? Okay, uh, now we want to show that this is going to be 1 to 1. Uh, so you could just look at the green line and say, Okay, well, this passes the horizontal line test. Um, but, I mean, I could have just drew this pretty poorly. I mean, just look at my handwriting. Uh, so what we can do instead is actually take the derivative of this and see if this is always increasing, are always decreasing. So prime of X is going to be so we use power rules that would be three times x minus one squared. And then we use chain will take the derivative of the inside, which is going to be one, um, and then this here will x minus one squared is always greater than 03 times that still greater than zero. So this is always greater than you go to zero, which implies FX is always increasing. So because it is always increasing, then we can say that function is 1 to 1 right now to get our domain and range for inverse. We don't even need to look at the graph of this. We just need to look at the domain and range of the functions that we, uh, started with over here. So the domain of F inverse is supposed to be the range of F. So that would just be negative. Infinity to infinity. And then the range of F in verse is supposed to be the domain of F, which is also negative infinity to infinity. So in this case, they have the same domain and range. Now, to find the inverse of this, we go home, this down, we can go ahead and first interchange f of X for y, and then we can interchange exit y. So this is going to be X is equal to why buy this one cubed. So first we take the cube root on each side. They cancel out over here. Um, and that would give just why minus one is equal to cube root of X, and then we just add one over. So why is going to be the cube root of X plus one? And so this is going to be our inverse. And now, to show that these compositions will give US X. Um, we just need to plug them in, so f inverse of f A bex. So this would be sort of taking a FedEx and plug it into our inverse. So it would be the Cube group of F of X plus one. We have the Q group of X minus one cubed plus one. So cube Cube counts while and then we just get That's my swan plus one. The ones are going to cancel, and we're just going to be left with X. Then we just need to flip this around. So the f of f inverse of X, which would be so f in verse. Yeah. Is, um so actually, what was African? Thanks much, one kid. So there's going to be f and birth defects minus one cubed. So we go ahead and plug in f inverse. So it be cubed root of X plus one minus one while cubed, the ones canceled hoped. Then we're left with Cuba. We have cube root of X cubed, and then that would just give us X. So again, you can see how this checks out as well

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