Why are non-linear activation functions required in multilayer perceptron classification? Solution: for some reason, I had forgotten that the non-linear activation function is applied at every layer of the neural network, not just at the output layer. Hopefully to others reading my original question below will understand why I asked it. Thank you for the answers, though.
Original: Suppose I have a multilayer perceptron network of a couple of layers with some output nodes that will be subject to the classic sigmoid activation function - how will this change which output node will have the highest value for a given input vector (and is selected as the final classification)? Namely, (denoting the sigmoid function as f(x)) if x' > x, f(x') > x, meaning the same output node will be selected as the final classification.
I think I am missing something about its importance in gradient descent or the determination of loss but please clarify this for me if you know my thinking error.

with your question, you can come back to the problem of XOR function that once was not solved by NNs without non-linear activation functions whereby NNs lost their attention until non-linear functions were involved.
https://stackoverflow.com/questions/33582251/can-an-ann-2-2-1-layers-be-implemented-to-learn-xor-using-linear-activation-fu
https://www.quora.com/Why-cant-the-XOR-problem-be-solved-by-a-one-layer-perceptron
