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# 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 2 x times the principal
square root of 5 minus 4. G of x is equal to x squared plus 2 x times the principal squared 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
sub, 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 its definition is right over there. And f of x, it's definition is right over
there. Let me do with another 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, g of x is x squared plus 2x. Times the principle 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 colour, 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, minus
4. And so now, we just simplify this. This is going to be equal to, we can get
rid of the parenthesis of this. x squared plus 2 x times the principle
square root of 5, minus 1. And now this is minus all of this, so we
can distribute the negative sign. You could do this as negative 1 times all
of this over here. So this is going to be equal to, if we
distribute onto that, we get. Minus 2x times the principle 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 over right
there. So, I'll just write 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. Over here we have a 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 the square root of, of
negative 5, minus 2 times x times the square root of
fi, sorry. 2 x times the square root of 5 minus 2 x
times square root of 5. That obviously cancels out. And then we're left with our constant
term. You have negative 1 plus 5. Add them together you get plus 4. And we're done. G minus f, 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.