Functions and graphs: describing dependence

The notation $f(x)$ means the value assigned by $f$ to input $x$. It does not ordinarily mean multiplication of $f$ by $x$. A function assigns exactly one output to every input in its domain. Different inputs may share an output: squaring sends both $2$ and $-2$ to $4$. This does not violate the function rule; it instead prevents an unrestricted inverse from assigning a unique original input.

A formula may impose restrictions even when no physical restrictions are present. For a real-valued square root, its argument must be nonnegative. Division requires a nonzero denominator. Applied models add further restrictions: a negative elapsed time might be mathematically accepted by a polynomial but outside the experiment. Always distinguish the formula's mathematical domain from the portion you intend to use.

The graph of a real function contains the points $(x,f(x))$ for allowed inputs. A horizontal coordinate names an input; the vertical coordinate records its output. A curve can be high while decreasing, or low while increasing. Height and change answer different questions. Keep axis labels and units visible, because the same shape can describe position, temperature, or an abstract score.

An affine rule has a constant output change per input change. Its intercept is the output at zero input, provided zero belongs to the domain. Its slope has output units divided by input units. People often call this a linear graph; in later linear algebra, “linear map” has the stricter requirement that zero goes to zero. Naming the distinction now prevents an avoidable surprise later.

Composition feeds the output of one function into another. In $g(f(x))$, evaluate $f$ first, then use its result as the input of $g$. The intermediate output must belong to the next function's domain. This is the mathematical version of a data pipeline, and it will reappear in neural networks when a layer's output becomes the following layer's input. Order usually matters.

A graph is helpful evidence but not a complete definition when only a finite window is visible. A sharp turn may disappear at low resolution, and a pole may lie just outside the displayed range. Use the algebraic rule to answer exact questions and the picture to build intuition. When a prediction is physically implausible, check whether you extrapolated beyond the domain where the rule was fitted or justified.