Reading inverse values from a table | Composite and inverse functions | Precalculus | Khan Academy

We're told the following table shows a few inputs and outputs of function g. All right, we have some possible inputs here for x and then the corresponding outputs here g of x. What is the value of g inverse of 54? So pause this video and see if you can figure that out before we work through it together.

All right, let's just remind ourselves what an inverse function even does. So if you have some value x and you input it into some function g, that function is going to output g of x. An inverse function takes us the other way. We could take this, what was the output of g, g of x? We can input that into an inverse function, the inverse function of g, and that is actually going to give us x. It's going to get us back to our original input right over here.

So what we're focused on right over here is g inverse of 54. So we can think about this part of this little chain that we set up. So what we're inputting into this inverse function is 54. So what we want to say is, all right, when g of x is equal to 54, what is x? And we can see that right over here; when g of x is 54, the corresponding input, original input, one way to think about it is 62. So this will be equal to 62.

Now, some of you might have been tempted to say, okay, look, it looks like I'm inputting a 54 into a function, so I'll say, okay, x is the input. Let me just go to 54 right over there as the input. But remember this 54 isn't an input into the inverse of g; this is an input into g of x. So if you wanted to evaluate this, if you wanted to evaluate g of 54, then you would look at the 54 up here and say, okay, that's going to be equal to 65. But we're looking at the inverse of g. So one way to think about it is when 54 is the output in g, what is going to be the input? And we see that that is 62.