Find the inverse function (on the given interval, if specified...

Question

Find the inverse function (on the given interval, if specified) and graph both $f$ and $f^{-1}$ on the same set of axes. Check your work by looking for the required symmetry in the graphs.

$f\left(x\right)=x^2+4$ , for $x\geq{0}$

Accepted Answer

Below there, today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Given the function F of X is equal to 3 x 2 + 2. For X is greater than or equal to 0, identify its inverse function from the following options, then graph the function and its inverse. Awesome. So it appears for this particular problem we're asked to solve for two separate answers. Our first answer, we're trying to figure out the inverse function of FF X. That is our first answer that we're trying to solve for, and one of our multiple choice answers indicated below will be our final answer for the inverse function. So that's our first answer. Our second answer is we're asked to create a graph that has the original function for F of X on it and a graph for the inverse function. So these two curves, the curve for the original function and the curve for the inverse function will be on the same graph, and that is our second and final answer we're trying to solve for. So we're trying to create a graph that has these two curves on it. So with that said, let us read off our multiple choice answers to see what our final answer might be, noting that our inverse function is denoted as F to the power of -1 of F of X is equal to some expression. So A is the square root of 2X minus 4. B is the square root of 2 X minus 6. C is the square root of X minus 3 divided by 2, and finally D is the square root of X minus 2 divided by 3. Awesome. So first off, in order to solve for this problem, let us write our function for F of X instead of saying F of X is equal to 3x2 + 2, let us simply write that Y is equal to 3 x 2 + 2. So now we need to switch X and Y. So when we switch X and Y for this expression, we will find that X is equal to 32 + 2. And now our our next step is we need to rearrange this equation to isolate and solve for Y because we need to solve for Y in terms of X. So when we rearrange this equation to. Isolate and solve for Y all by itself using a little bit of algebra. I'm gonna skip some steps because you should be able to rearrange this equation to isolate and solve for Y on your own. That's just simply algebra. So when you do that, you will find that Y. Is equal to. The square root of X minus 2 divided by 3. So this is all under the same square root symbol. So Y is equal to the square root of X minus 2 divided by 3. So therefore, without a doubt, the inverse function, which will be F to the power of -1 of X will be equal to the square root of X minus 2 divided by 3, and that is our final answer. For the inverse function hooray, we did it. And this corresponds with multiple choice answer letter D. Which once again will be that after the power of -1 of X is equal to the square root of X minus 2 divided by 3. But we're not quite done solving our entire problem yet because we're also asked to make a graph of the original function and the inverse function. So we now need to switch gears and start towards working towards creating a graph. For these two equations, or rather these two functions. So now in order to graph, we need to create or not create rather the proper word or proper phrasing would be we have to figure out coordinate points for both of these curves because we need to figure out at least minimum two coordinate points for each function in order to connect them to create a curve. For these two functions to plot on our graph. So with that said, Let us start with our original function first. So let us focus our attention on F of X is equal to 3 X2 + 2. Awesome. So now what we're trying to do is we need to make sure that we are focusing on our range because our range is stated to be X is greater than or equal to 0. So we need to plug in different values for X to figure out our coordinate points. So let us start using our equation for the original function. Let us state that X is equal to 0 because we know that X is greater than or equal to 0. So when X is equal to 0. That will mean that F of 0, because you're plugging in 0 for X, so F is 0 will be equal to. 3, so we need to take 3. So 3 multiplied by, so this is gonna be 3 multiplied by 0 to the power 2 plus 2. So that will mean when we plug that into our calculator, our final answer will be 2. And similarly, for X is equal to 1 because 1 is greater than 0, that will mean that F of 1, so instead of having to write the equation out again, I'm just going to give you the answer. So instead of plugging in 0, you're gonna plug in 1, and when you do that, you will get that F of 1 is equal to 5, and I said 2 points minimum, but since this is Not a line, so for when you're plotting any sort of line curve, you only need two coordinate points, but since this one's like more of a curvy line, since it has like a, if you really want to properly graph the curve, you're gonna need 3 coordinate points minimum. So let's do one more. So now let's say that X equals 2 because 2 is greater than 0. So when F, So we need to note that F of 2. So when you plug in 2 for X, that will mean that it's gonna be equal to 14. Fantastic. We are on a roll here. So that is how we create our coordinate points for our first function, which ultimately we should note that our three coordinate points will be 0.2, noting that this is our XY. like format, so this is 0.2 is our first point. Then our second point is 1.5, and finally, our last coordinate point is 2.14. Fantastic. So that's our XY coordinate points. Fantastic. So essentially you're going to be taking these three coordinate points, and you're going to be taking them, plotting them on the graph, and then connecting them together with a curve. Awesome. So now, that's our original function. Now we need to figure out the coordinate points for our inverse function. So let us. Figure that out now. So now we're focusing on Once again, F of the power of -1 of X, so this is our inverse function, which we're gonna say is equal to Y and Y is equal to the square root of. X minus 2 divided by 3. So Y is equal to the square root of X minus 2 divided by 3. Awesome, so we're gonna take this expression. And we're gonna do that, we're gonna follow the same exact steps as we did for our original function. So we're going to be plugging in different values for X into our equation for Y in order to figure out our three separate coordinate points. So once again, considering our range, let us say for our first condition X is equal to 2. So when X is equal to 2, that will mean that Y is equal to the square root of 2 minus 2 divided by 3. So that will mean that our first answer for Y is 0. And for our second condition, let's say that X is equal to 5. So when X is equal to 5, Y is going to be equal to 1, and instead of plugging in. Once again, let's be specific here. set up plugging in 2, like we did for X equals 2, we are plugging in X equals 5, and when we do, we will get that Y is equal to 1. And similarly, for our last quarter point, let's say that X is equal to 9, and when X is equal to 9, that will mean that Y is approximately equal to 1.53. Fantastic. So that will mean our coordinate points should be 2.0. And our next corner point should be 5.1, and our last coordinate point should be 9.1.53. So once again, we need to plot these 3 coordinate points onto a graph and then connect them with a curve. And that is how we're going to plot that specific curve for our inverse. So with that in mind, our next step now is we need to take our three separate coordinate points for both the original function and the inverse function and then plot them on a graph. So I've gone ahead. And plugged in our original function and inverse function into some into a graphing software and I graphed all, so I graphed it as three separate graphs. So we did, I broke up to the curves for our original function. So going from the far left working our way to the right. So we have our original function which is for Y equals F of X, and as you can see when we plot our 3 points, we got 2. 0. And then we have of course our third point, which is 2.14, and then our second point, which is 1.5. So we plotted our three coordinate points for Y equals F of X. And then moving to the right, we have our graph for Y equals our inverse function, which is F-1 X. And like we mentioned below, we have our three separate coordinate points. Once again, so we don't, so not to get repetitive here, we plotted our 3 separate coordinate points and as we could see we have it at 0.2. As you can see, or 2 rather, it should be because you go over to up 0, so 2.0, and then we have it at 51, and then at 9.1.53. So with that said, so now there's two separate ways you could do this particular problem for the graph. So you could do them separate. You could graph each of these curves separate for the original function and the inverse function as two separate graphs, and then combine them, or you can just combine them onto the same. Graph with the same pair of axis on right from the beginning. It's all up to you. It doesn't matter. But as long as for your final answer, you need to note for your final graph that they have to be both curves for both the original function and the inverse function have to be on the same graph. And that is our final answer. Hooray, we did it. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Find the inverse function (on the given interval, if specified) and graph both $f$ and $f^{-1}$ on the same set of axes. Check your work by looking for the required symmetry in the graphs.
$f\left(x\right)=x^2+4$ , for $x\geq{0}$

Key Concepts

Inverse Functions

Graphing Functions

Quadratic Functions

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