f(x) = square root of (x^2 - 10x - 11)find the largest possible domain

To find the largewst possible domain of f(x) =
sqrt(x^2-10x-11)

The domain of f(x) = sqrt(x^2-10x-11) is
the set of all possible values for which f(x) is real.

The
right sqrt (x^2-10x-11) is not real when (x^2-10x-11) < 0 (or
negative).

sqrt(x^2-10x-11) real only when (x^2-10x-11)
> 0.

Now we factorise
x^2-10x-11.

x^2-10x-11= x^2-11x
+x-11

x^2-11x+x -11 =
x(x-11)+1(x-11)

(x-11)(x+1)
.

Therefore x^-10x-11 = (x+1)(x-11) could be > 0
only when both factors (x+1) and (x-11) are negative or both factors (x+1) and
(x-11)   are posive.

So this is possible only when x
< -1  or x > 11.

Therefore the largest domain
of x for f(x) to be real is x should belong to the interval  or domain (-infinity , -1)
U (11 , +infinity).