Fundamental Examples

Same formula, different function

The expression $x^2$ can define several different functions:

• $f:\mathbb{R}\to \mathbb{R}$, $f(x)=x^2$ is not injective.
• $g:[0,\infty )\to [0,\infty )$, $g(x)=x^2$ is bijective.
• $h:\mathbb{C}\to \mathbb{C}$, $h(z)=z^2$ lives in a different number system.

The formula alone is not the whole function. The domain and codomain are part of the object.

Continuous growth

If a quantity grows at a rate proportional to itself, a simple model is $N(t)=N_0{e}^{kt}.$ Here $t$ is often time, $N_0$ is the initial value, and $k$ is the growth/decay constant. If $k<0$, the same form describes decay. This appears in populations, radioactive decay, capacitor discharge, and many linearised physical systems.

Composition

If $f(x)=x^2$ and $g(x)=\sin x$, then $(g\circ f)(x)=\sin (x^2),(f\circ g)(x)={\sin }^{2}x.$ Composition usually does not commute. Check order carefully, especially in transformations, units conversions, and ML pipelines.

Domain check

The rule $q(x)=1/(x-3)$ is undefined at $x=3$. A valid real domain is $\mathbb{R}\setminus \{3\}$, not all of $\mathbb{R}$.

Example: why domains matter

The formula

f(x)=rac{1}{x}

looks simple, but it is not a function $\mathbb{R}o\mathbb{R}$ because $x=0$ is forbidden. A correct version is

f:\mathbb{R}\setminus\{0\} o\mathbb{R},\qquad f(x)=rac{1}{x}.

This is the kind of small precision that prevents later calculus disasters.