In Exercises 39-52,
a. Find an equation for ƒ¯¹(x).
b. Graph ƒ an...

Question

In Exercises 39-52,
a. Find an equation for ƒ¯¹(x).
b. Graph ƒ and ƒ¯¹(x) in the same rectangular coordinate system.
c. Use interval notation to give the domain and the range off and ƒ¯¹. f(x) = (x+2)³

Accepted Answer

Welcome back. I am so glad you're here. We are given the function F. Of X, which equals X plus three in parentheses raised to the third power. And we need to do several things here, we need to find the equation for its inverse which is going to be f inverse of X equals something. We need to graph both functions. So I'm going to draw a quick coordinate plane with a vertical Y axis, the horizontal X axis. They meet together at the origin in the middle and then we're not going to need to identify the domain and range in interval notation. So let's get started and we'll get started by figuring out the equation for the inverse. So I'm going to rewrite the function F of X down here, F of X equals X plus three. Cute. And then I'm going to rewrite the F of X part as Y. So Y equals X plus three cubed. And then we'll enter into the twilight zone a little bit. And wherever there's a Y will substitute an X. So instead of Y equals X equals and wherever there's an X will substitute a Y. So instead of X plus three it's Y plus three. And that's all cubed. And now what we need to do is get that why all by itself? Because why is just miserable and forlorn until it is by itself on one side. So to do that, the first step is we're going to take the cute of both sides. So on the left will have the cubed root of X and that's equal to Y plus three. Now to get that Y ever more by itself will subtract three from both sides. So the left we have the cubed root of x minus three equals Y. Y. Is thrilled. It is celebrating it is throwing a party for being all by itself. And this cubed root of X -3 that is our inverse. So we plug it back in here to the equation the cubed root of X. And then outside the radical -3 There is our equation for the Inverse. Alright one step done. Now let's work on graphing. Okay so if you know what X plus three cute looks like. That's great if you don't you can choose some points and plot them and you first want to look and see if you have any domain restrictions while with cubing nope we don't have any domain restrictions and then you want to choose some X values, plug those in to the original function, get some Y values as a result and then plot those points for X values. I'm going to choose zero negative three and negative six. And that's because I've done this problem before and tried a lot of points and was frustrated by all the different points. I tried so I'm going to use these three and now plugging those in for X. So we've got instead of X plus three. Zero plus three cubed zero plus three is three. So we've got three cubed and three cubed equals 27. So starting at the origin we're not going left or right for our X value. We're going up 27 units for our Y. Value. And next we're plotting negative three for X values and negative three plus three cute equals and negative three plus three is zero. So we've got zero cute and zero cubed 20. So from the origin will go to the left three units for our X. Value and nowhere for our Y value. Neither up nor down. So I'm going to put an X intercept at negative 30 for our F. Of X. And now for negative six. Plugging that in for X negative six plus three cubed negative six plus three is negative three and negative three cubed is negative 27. So we'll start at the origin will go to the left six and then we'll go down negative 27 units. And that looks like it could be a straight line but if you recall from previous lessons this cubed anything cubed gets a little bit curvy, more curvy that you can use your imagination. But that's going to do for the F. Of X. It's rough but it'll allow us to find the appropriate answer choice. If you needed to have a really precise one. You would just need to plot some of these numbers in between zero and three and in between negative 30 negative three and negative three and negative six. Okay there's my horribly drawn affects. Now let's figure out our inverse. Okay. Before we set up our inverse, as I said, we need to check our domain and our domain for our inverse is going to be related to the domain of our original one. So the domain for the original function all the possible X values. Well, it looks like it could be absolutely anything from negative infinity to positive infinity. And we put that in interval notation with parentheses negative infinity comma infinity parentheses. Now for the range, those are all the possible why values so looks like it could be anything from negative infinity to positive infinity again. So our range is again parentheses negative infinity comma infinity parentheses. Okay, we've got the domain and range. And now here's how our inverse is related to that. The domain where are inverse is the range of our original function. So the domain here is negative infinity to infinity. So no domain restrictions. And while we're at it we can state what the range for our inverse function is. The range is the domain of our original function which is negative infinity to infinity. Ok. We've got the domain and range identified. Now let's graph are inverse. Okay, now you may recall from previous lessons that the inverse of a function is just going to be switching around the X. And the Y values. So let me show you how this would work if we chose X values of 27. 0 and negative 27. Those are the Y values from the original function. Let's plug those into the equation and see what we get for our Y values here. So I'm going to plug 27 in for X. the cubed rut of 27 minus three. So the cubed root of 27 that's 33 minus three. That equals zero. All right now let's plug zero in. So we've got the cubed root of zero minus three. Well the cubed root of zero is zero. So that's zero minus 30 minus three is negative three. Ha look at that. Alright and now let's plug in negative 27 we have the cubed root of negative 27 minus three. The cube of negative 27 is negative three, negative three minus three is negative six. Check that out. Alright. We could plot these points. Looking at our coordinate planes starting at the origin. We are going to the right 27 for our X. Value and we're not going up or down for our Y value. So that's somewhere like here and now for our next one we've got zero for our X. Value and down three for our Y value. So that's about here this is the y intercept of zero negative three. And for our last point we've got negative 27 from the origin to the left 27 for our X value and then down six for our Y value. So that's here ish. Now this is going to be even less accurate for my squiggle ease. I'm sure maybe it's better. That kind of looks a little better. Okay, there's what are F inverse of X is doing okay. We have got it graft. We have got the equation for the inverse. We have got the domain and the range. We have done it all. Now let's identify the correct answer choice and we will kind of zoom in on looking for an ffx value that has an X intercept at negative 30. So looking at answer choice A F of X has an X intercept at negative 30. Cool. Alright, let's look at our inverse. R F inverse has a Y intercept at zero negative three. And yeah, the Y intercept is at zero negative three. Cool. Cool. Alright. A has got potential. Let's check the other ones real quick. So for B that has an X intercept at 30, nope. We wanted negative 30. Sorry, B. You are just rejected. All right, come on the journey. I don't know if that whole thing is going to make it but come on the journey let's check out C and E. No, maybe maybe we got the whole thing. Oh, missed part of an arrow. Okay. See Andy wow, those look kind of different. Okay. We're looking for an F of X. X intercept at negative 30 and C. S X intercept is at 60. Sorry, that is rejected and for D F of X has an X intercept negative 60, nope. Sorry, you are rejected. Okay. Let's go back up to our beautiful, lovely answer choice A that just cooperated so well and seemed so great. And let's check all the all the parts because we did a lot of work here. Alright, so we've got a formula of f inverse of x equals the cubed root of X -3. Yes, that matches the domain of both F of X and F of F inverse of X is negative infinity to positive infinity. Yes, those domains work. And the range of both functions is negative infinity to positive infinity. Yes, that works. Our graph, I mean their graph is much better, but the graphs generally match. So it looks like answer choice A. You've got it. Well done. We'll catch you on the next one.

In Exercises 39-52, a. Find an equation for ƒ¯¹(x). b. Graph ƒ and ƒ¯¹(x) in the same rectangular coordinate system. c. Use interval notation to give the domain and the range off and ƒ¯¹. f(x) = (x+2)³

Key Concepts

Inverse Functions

Graphing Functions

Domain and Range

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