Input-Output Table
An input-output table shows how a rule or function transforms input values into output values, with each input paired with exactly one output.
Definition
An input-output table shows what comes out when you apply a rule to different inputs: each input has exactly one output, like feeding a number into a machine and getting one specific number back out. It organizes values of an independent variable (input/$x$) alongside the corresponding dependent variable (output/$y$) under a consistent rule, and identifying that rule from the table, by noticing that constant differences indicate a linear rule while constant second differences indicate a quadratic one, is the process of finding the pattern or function. Formally, an input-output table is a finite representation of a function $f: X \to Y$, listing $(x_i, f(x_i))$ for a selected subset of the domain; in computer science such a table corresponds to a lookup table or truth table, and determining the function from a finite sample is an inverse problem that is generally underdetermined without additional constraints.
Example
Rule: "multiply by $3$ and add $1$." Input $2$ gives output $7$; input $5$ gives output $16$; input $0$ gives output $1$. Given inputs $1$, $2$, $3$, $4$ and outputs $5$, $8$, $11$, $14$, the differences between consecutive outputs are all $3$, suggesting the rule $y = 3x + 2$, which checks out since $3(1) + 2 = 5$ and $3(2) + 2 = 8$. Given $4$ input-output pairs of an unknown polynomial, at most a degree-$3$ polynomial is uniquely determined by Lagrange interpolation; additional pairs either confirm the polynomial or reveal higher-degree structure.
Key Insight
Think of the rule as a machine: you feed a number in, the machine does its thing, and one specific number comes out. Constant differences in an output column indicate a linear rule, and constant second differences indicate a quadratic one, patterns that speed up equation identification. Overfitting in machine learning is the modern analog of choosing a high-degree polynomial to fit a small table perfectly; preferring the simplest explanation that fits the data, lowest degree or fewest parameters, is a statistical form of Occam's Razor.