I'm a bit confused. Are you trying to find the value of a such that f_a is it's own inverse? And I'm not sure how you got your final equation.

If you are trying to find an a such that f_a is its own inverse:

Clear the complex fractions:

Now put the whole equation under a common denominator, cancel the denominator, and this leaves you a polynomial in powers of x. You will see that the x^4 coefficient cannot be set to 0, so this problem cannot be solved for any value of a.

-Dan