This confirms that an MLP, despite its quite convoluted mathematical form, is nothing more than an analytic function, specifically a mapping of real-valued vectors \( \hat{x} \in \mathbb{R}^n \rightarrow \hat{y} \in \mathbb{R}^m \).
Furthermore, the flexibility and universality of an MLP can be illustrated by realizing that the expression is essentially a nested sum of scaled activation functions of the form
$$ \begin{equation} f(x) = c_1 f(c_2 x + c_3) + c_4 \tag{10} \end{equation} $$where the parameters \( c_i \) are weights and biases. By adjusting these parameters, the activation functions can be shifted up and down or left and right, change slope or be rescaled which is the key to the flexibility of a neural network.