Functions are machines

A rule with one output

A function assigns exactly one output to every input in its domain. With $g(x)=2x$, the input 3 produces 6; the input $-2$ produces $-4$. Different inputs can share an output: $f(x)=x^2$ sends both 2 and $-2$ to 4. That still counts as a function.

Write $g:\mathbb R\to\mathbb R$ to say that $g$ accepts real inputs and returns real outputs. The set after the arrow is the codomain. The outputs actually reached form the range. For $f(x)=x^2$ with real inputs, the range is $[0,\infty)$ even if we choose $\mathbb R$ as the codomain.

Connect two simple rules

Let $g$ double a number and let $f$ square a number. Connect the output of $g$ to the input of $f$:

The composite function is written $f\circ g$. Read it as “f after g.” Evaluate the inside first:

Changing the order usually changes the answer:

For $x=2$, those outputs are 16 and 8. Composition is an instruction sequence, not ordinary multiplication of function names.

Ask where a function makes sense

A formula comes with a domain. The real-valued square root $r(x)=\sqrt{x}$ needs $x\ge0$. A reciprocal $h(x)=1/x$ excludes zero. The natural logarithm needs $x>0$. Composing functions means checking the intermediate output too: $\ln(x-2)$ requires $x>2$.

A function may accept several numbers. For example,

could be the cost of $x$ notebooks and $y$ pens, at 2 dollars and 3 dollars each. This function has a pair as its input and one number as its output.

Reversing a rule

An inverse recovers the input from the output. The inverse of $g(x)=2x$ is $g^{-1}(u)=u/2$. The superscript $-1$ here means inverse function, not reciprocal.

Squaring on all of $\mathbb R$ has no inverse function: from the output 4, you cannot tell whether the input was 2 or $-2$. Restrict the input to $[0,\infty)$ and the principal square root becomes an inverse.

An ML connection

A prediction pipeline is a sequence of transformations. You might first rescale an input, then compute a score, then map that score to a prediction. Understanding the order now will make the chain rule easier later.