You are correct. It is possible to construct a linear neural network, where an algebraic function could be extracted, but in practice, almost all networks use non-linear activation functions.

In the case of a linear network, the function would be a dot product between the input and the weights: 𝑥1𝑤1+𝑥2𝑤2+𝑥3𝑤3 ... for all inputs (xi) and weights (wi)