If f(x)= 3x+6, prove that f( f(x)) is an increasing function.

To establish that a function is increasing, we'll have to
do the first derivative test. To do the first derivative test, we'll have to determine
the result of the composition of f(x) with f(x).

f(x)*f(x)
= f(f(x))

We'll substitute in the expression of f(x), the
variable x by the expression of f(x).

f(f(x)) = 3f(x) +
6

f(f(x)) = 3(3x+6) + 6

We'll
remove the brackets:

f(f(x)) = 9x + 18 +
6

We'll combine like
terms:

f(f(x)) = 9x + 24

Since
we know the expression of f(f(x)), we can do the first derivative
test.

f'(f(x)) = (9x +
24)'

f'(f(x)) = 9

If the first
derivative is positive, then the original function is
increasing.

Since the result of the first
derivative test is positive, then f(f(x)) is an increasing
function.