Question 1010711

I am stuck on this problem.  If I can see this worked out it would help a lot!

If f(x) = (1/x - 9x) / (1+3x), what is f(1/x).

My book says the answer is x-3 but I cannot get this same answer.
<pre>{{{f(x) = (1/x - 9x)/(1 + 3x)}}}
{{{f(1/x) = (1/(1/x) - 9(1/x)))}}}{{{"÷"}}}{{{(1 + 3(1/x))}}}
{{{f(1/x) = (1 * (x/1) - 9/x))}}}{{{"÷"}}}{{{(1 + 3/x)}}}
{{{f(1/x) = (x - 9/x))}}}{{{"÷"}}}{{{(1 + 3/x)}}}
{{{f(1/x) = ((x^2 - 9)/x)}}}{{{"÷"}}}{{{((x + 3)/x)}}}
{{{f(1/x) = ((x - 3)(x + 3)/x)}}}{{{"÷"}}}{{{((x + 3)/x)}}} ------ Factoring {{{x^2 - 9}}}
{{{f(1/x) = ((x - 3)(x + 3)/x)}}}{{{"*"}}}{{{(x/(x + 3))}}} ------ Applying KEEP, CHANGE, FLIP
{{{f(1/x) = (highlight((x - 3))cross((x + 3))/cross(x))}}}{{{"*"}}}{{{(cross(x)/cross((x + 3)))}}}
{{{highlight_green(f(1/x) = x - 3)}}}