Designing a discrete-time inverse system
Correct answer is in the comment already but for completion: Your transfer function is 

$$H(z) = b_0 + b_1 \cdot z^{-1}, \mbox{ with } b_0 = 1, b_1 = .9$$
which makes the inverse
$$H^{-1}(z) = \frac{1}{b_0 + b_1 \cdot z^{-1}} $$
which corresponds to the difference equation
$$y(n) + 0.9 \cdot y(n-1) = x(n) $$ or
$$y(n) = x(n) -0.9 \cdot y(n-1)  $$