$f\left(x\right)=-x$  $\left\{x\in \mathrm{ℝ}\right\}$

\begin{align*} Let\quad f(x) & =y\\ y & = -x\\ -y & =x\\ Therefore\\ f^{-1}(x) & = -x \end{align*}

$f\left(x\right)=\frac{1}{x}$  $\left\{x\in R,x\ne 0\right\}$

\begin{align*} Let\quad f(x) & =y\\ y & = \frac{1}{x}\\ xy & =1\\ x & =\frac{1}{y}\\ Therefore\\ f^{-1}(x) & = \frac{1}{x} \end{align*}

$f\left(x\right)=-\frac{1}{9x}$ $\left\{x\in R,x\ne 0\right\}$

\begin{align*} Let\quad f(x) & =y \\ y & = \frac{-1}{9x}\\ 9xy & =-1\\ x & = \frac{-1}{9y}\\ Therefore\\ f^{-1}(x) & = \frac{-1}{9x} \end{align*}