Conceptual Bottlenecks Math Example 4

Solution

1. 1
   With numbers, objects are static values like $3$ or $\pi$. With functions as objects, we treat the whole rule $x \mapsto f(x)$ as a single entity.
2. 2
   This shift is required for: derivatives (the derivative of $f$ is a new function $f'$), function composition ($f \circ g$ is a new function), and higher-order functions (a function that takes a function as input).
3. 3
   Example: integration as a linear operator $I(f) = \int_0^1 f(x)\,dx$ takes a function $f$ as input and returns a number. Without thinking of functions as objects, this concept is inaccessible.