Multivariate calculus from first principles
So far the output was a single number. Let it grow into a vector too. A function f: Rⁿ → Rᵐ takes a vector in and returns a vector out: many numbers in, many numbers out. That is the exact shape of a neural-network layer, where an input vector enters and a transformed vector leaves.
The way to understand any vector-valued function is to read it one output coordinate at a time. Each output component is itself an ordinary scalar function Rⁿ → R, called a component function. Stack m of them and you have the whole map.
A mixing desk turns a few input dials into several output readings at once: nudge the sliders and every meter responds together. That is a function f: Rⁿ → Rᵐ: a vector of inputs goes in, a vector of outputs comes out. To understand it you read one meter at a time, since each output coordinate f₁, f₂, and so on is its own ordinary recipe built from the same input dials.