Difference of functions | Functions and their graphs | Algebra II | Khan Academy

We're told that f of x is equal to two x times the square root of five minus four, and we're also told that g of x is equal to x squared plus two x times the square root of five minus one. They want us to find g minus f of x, so pause this video and see if you can work through that on your own.

The key here is to just realize what this notation means: g minus f of x is the same thing as g of x minus f of x. And so again, if this was helpful to you, once again I encourage you to pause the video.

All right, now let's work through this again. This is going, or I guess the first time, but now that we know that this is equal to g of x minus f of x, so what is g of x? Well, that's the same thing as x squared plus 2x times the square root of 5 minus 1.

And what is f of x? Well, it's going to be 2 x times the square root of 5 minus 4. We are subtracting f of x from g of x, so let's subtract f of x from g of x. Now it's just going to be a little bit of algebraic simplification.

This is going to be equal to x squared plus 2x times the square root of 5 minus 1. Now we just have to distribute this negative sign. So negative 1 times 2x times the square root of 5 is going to give us minus 2x times the square root of 5.

Then the negative of negative 4 is positive 4. Let's see if we can simplify this sum. We only have one x squared term, so that's that one there. So we have x squared, and let's say we have 2x times the square root of 5, and then we have another, oh, now we subtract 2x times the square root of 5.

So these two cancel out with each other, so those cancel out, and then we have minus 1 plus 4. If we have negative 1 and then we add 4 to it, we're going to have positive 3.

So if we just factor this and take this into consideration, four minus one is going to be equal to three. And we're done! That's what g minus f of x is equal to: x squared plus three.