# Subtracting functions

Given that f(x)=2x√5-4 and g(x)=x^2+2x√5-1, Sal finds (g-f)(x). Created by Sal Khan and Monterey Institute for Technology and Education.
Video transcript
f of x is equal to 2x times the principal square root of 5 minus 4. g of x is equal to x squared plus 2x times the principal square root of 5 minus 1. Find g minus f of x. And once again, g minus f of x is really just referring to a function that's made by subtracting the function definition for f from the function definition of g. So this is going to be the same thing as g of x minus f of x And then we can use the function definition to substitute back here to figure out what g minus f of x actually is. So we see g of x. g of x, it's definition is right over there. And f of x, it's definition is right over there. Let me do the color the contrast with that pink better. So f of x is right over there. So g of x minus f of x is going to be equal to g of x is x squared plus 2x times the principal root of 5 minus 1. And then from that-- and I can even put parentheses around it. From that we are going to subtract-- let me do the subtraction in that yellow color. From that, we are going to subtract f of x. And f of x is given right up here, is 2x times the principal square root of 5 minus 4. And so now we just simplify this. This is going to be equal to-- we can get rid of the parentheses of these. x squared plus 2x times the principal square root of 5 minus 1. And now this is minus all of this. So we can distribute the negative sign. You could view this as negative 1 times all of this over here. So this is going to be equal to-- if we distribute on to that, we get minus 2x times the principal square root of 5. Negative 1 times negative 4 is plus or positive 5. And now we can simplify. We have one second degree term right over there. So I'll just write it down, x squared. We have two terms that deal with x times the square root of 5. We have this one right over here, 2x times the square root of 5. And then over here, we have minus 2x times the square root of 5 So we have something, and then we subtract that same something. So these two are going to actually cancel out. Those two are going to cancel out. 2 of x times square root of negative 5 minus 2 times x times the square root of 5-- Sorry. 2x times the square root of 5 minus 2x times the square root of five. That obviously cancels out. And then we're left with our constant terms. We have negative 1 plus 5. Add them together, you get plus 4. And we're done. g minus f of x is equal to x squared plus 4, which is the difference of these two function definitions. It's g of x minus f of x.