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# Adding functions

Given that f(x)=9-x^2 and g(x)=5x^2+2x+1, Sal finds (f+g)(x). Created by Sal Khan and Monterey Institute for Technology and Education.

Video transcript

f of x is equal to 9 minus x. Actually, it should
be 9 minus x squared. And g of x is equal to 5x
squared plus 2x plus 1. And then they say
find f plus g of x. And this looks a
little bit bizarre. What kind of notation is this? And that's really the
core of this problem is just to realize that
when someone writes f plus g in parentheses like
this of x this is just notation for-- so f plus g of
x, I just rewrote it. This is the same thing. This is just a kind
of shorthand notation for f of x plus g of x. You could view f plus g
as a new function that's created by adding the
other two functions. But when you view it
like this-- so this is really what we have to find. Then, you just have to
add these two functions. So f of x, they've given the
definition right over there, is 9 minus x squared. And g of x, they've given the
definition right over here, is 5x squared plus 2x plus 1. So when you add
f of x to g of x, this is going to be
equal to-- and I'm just rewriting a lot of things
just to make it clear. The f of x part is
9 minus x squared. And then you have the plus. I'll do that in yellow. Plus the g of x part, which
is 5x squared plus 2x plus 1. And now we can just
simplify this a little bit. We can say-- let me just
get rid of the parentheses. I'll just rewrite it. So this is equal to
9 minus x squared plus-- since this
is a positive, we don't have to worry
about the parentheses. So plus 5x squared
plus 2x plus 1. And then we have two x squared
terms or second degree terms. We have 5x squared. And then we have
negative x squared. 5x squared minus x
squared is 4x squared. So you get 4x squared when
you combine these two terms. You only have one first degree
term 2x there, so plus 2x. And then you have two constant
terms, or 0 degree terms. So you have the 9
and you have the 1. And 9 plus 1 gets us to 10. So we're done. f plus g of x is equal to
4x squared plus 2x plus 10. Notice this is a
new function that's created by summing the function
definitions of f and g.