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In a Gaussian Process (GP), we know that choice of the covariance function determines the shape of function that can be drawn from the GP.

eg.

Constant : $\sigma _{o}^{2}$
Draws constant function based on the hyper parameter $\sigma_{o}$

Linear kernel : $k_{\textrm{Lin}}(x, x') = (x - c)(x' - c)$
Draws linear functions based on the hyper parameter $c$

Is there a covariance function that draws inverse function eg. $f(x) = 1/x$

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I think you misunderstood what the covariance function does. It quantifies the correlation between any two outputs $f(\mathbf{x}), f(\mathbf{x}')$ as function of the inputs $\mathbf{x},\mathbf{x}'$. The functions are still random and by changing the kernel, compare for example the RBF and the Exponential kernel, the function gets from smooth to wiggly. Therefore my answer is no, there is not.

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