Q.If $f(m)=m^2-3m+1$, find $f(x+1)$.

A function $f$ from a set $A$ to a set $B$ (written $f:A\to B$) is a rule that associates to every element $x\in A$ a unique element $y\in B$, written $y=f(x)$. Two things can break this: an element of $A$ left with no image at all, or an element of $A$ given two different images — either failure means the relation is not a function.

$A$ is called the domain (the full set of allowed inputs), $B$ is the co-domain (the set outputs are guaranteed to land inside), and the range is the smaller set $f(A)=\{y\in B\mid y=f(x)\text{ for some }x\in A\}$ of values actually produced — always a subset of the co-domain, though it may equal it. A function can be shown in six equivalent ways: a verbal description, an arrow diagram between two sets, a set of ordered pairs, an algebraic rule/formula, a table of values, or a graph — with domain and range read off each form appropriately (e.g. on a graph, domain is the horizontal spread/projection onto the $x$-axis, and range the vertical spread/projection onto the $y$-axis). …