Is it possible to predict $x^2$, $\log(x)$, or variable function of $x$ using RNN? There were some posts that using RNN can predict the next point of the sine wave function with data history.
However, I wondered if it also works on all the functions of $x$, such as $x^2$, $x^3$, $\log(x)$, $\frac{1}{(x+1)}$ functions.
Yes, RNN can work on the functions you have mentioned. In fact, neural networks can approximate anything (Universal Approximation Theorem). This question also reminds me of Neural Turing Machine.
But, it would be a complete waste to use RNNs or NNs for such a task.