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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 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.