Let f(x)= 12-4x and g(x)= 1/x.......what is the domain of g(f (x))

We'll determine the result of the composition of the
functions:

(gof)(x) = g(f(x))

That
means that in the expression of g(x), we'll substitute x by
f(x):

g(f(x)) = 1/f(x)

We'll re-write
the result putting the expression of f(x) at denominator of the composed
function:

1/f(x) = 1/(12-4x)

Now, we'll
impose the constraint that for the ratio to exist, the denominator has to be different from
zero.

More accurate, we'll compute the value of x that makes the
denominator to cancel and we'll reject them form the domain of the function
g(f(x)).

12 - 4x = 0

We'll subtract 12
both sides:

-4x = -12

x =
-12/-4

x = 3

So, the
domain of definition of the composed function is the real set R, without the value of
3.

x belongs to the interval (-infinite
; +infinite) - {3}