Find the inverse of the function 𝑓 𝑥 is equal to six 𝑥 cubed.
In order for us to find the inverse of this function, I want to first sketch a curve just to help us understand actually what the inverse of a function actually is. So here’s our rough sketch of our function. And if we actually want to find the inverse of this function, what it actually means is a reflection around the line 𝑦 equals 𝑥.
And by doing this, you can actually see that our 𝑥- and our 𝑦-coordinates will actually be switched. I know this is only a rough sketch. If we have a look at these two points, you can see that actually the 𝑥- and 𝑦-coordinates would be switched.
Okay, so now we have actually have an understanding of what the inverse is, we can use that to solve the problem. So then if we say that 𝑦 is equal to 𝑓𝑥, which is equal to six 𝑥 cubed, so therefore what we want to do is to first of all find the inverse. And in order for us to do that, what we’re gonna to do first is we’re going to exchange the variables. So therefore, we get that 𝑥 is equal to six 𝑦 cubed. Now, what we actually want to do to find our inverse is that we want to express 𝑦 in terms of 𝑥. And to enable us to do that, we need to rearrange.
The first step was to divide both sides through by six, which gives us 𝑥 over six is equal to 𝑦 cubed. And then, we’ve taken the cube root of both sides, which gives us cube root of 𝑥 over six is equal to 𝑦. And therefore, we can say that the inverse of our function is equal to the cube root of 𝑥 over six. And this is actually the case because we exchange the variables.
So we can now just quickly go through the steps again. Step one, you need to exchange the variables. So we need to swap 𝑦 for 𝑥. Then, we rearrange to express 𝑦 in terms of 𝑥. And therefore, you do that and you obtain 𝑦. And the 𝑦 is gonna be the value that is the inverse of your function.