Verify if f(g(x))>=0 for f(x)=x^2+2x+1; g(x)=x-2011

f(x)= x^2 + 2x + 1

g(x) = x-
2011

We need to verify if fog(x) is zero or
positive.

Let us determine the function fog(x) =
f(g(x))

==> f(g(x))= f(x-2011) = (x-2011)^2 +
2(x-2011)+1

==> f(g(x)) = x^2 - 4022x + 2011^2 + 2x
- 4022 + 1

==> f(g(x))= x^2 - 4020x +
4040100

==> f(g(x))=
(x-2010)^2

We notice that (x-2010)^2 is always positive or
zero when x=2010

Now we proved that f(g(x) > 0 for
all R-{ 2010}

Also f(g(x))= = 0 for x =
2010.

Then f(g(x)) >= 0