Q.The rule $f(x)=x^2$ is a bijection if the domain and the co-domain are given by

Definition. A relation $f\subseteq A\times B$ is a function $f:A\to B$ if (i) every $a\in A$ has some image $b\in B$ with $(a,b)\in f$, and (ii) that image is unique: $(a,b),(a,c)\in f\Rightarrow b=c$. Then $f(a)=b$; $b$ is the image of $a$, $a$ is a pre-image of $b$. The range $\{b:(a,b)\in f\text{ for some }a\}$ is always $\subseteq$ co-domain $B$. Only the domain side is required to be fully, uniquely covered -- how many pre-images a co-domain point has, or whether it has any, are separate questions (injectivity/surjectivity below).

Representing a function: tabularly (a list of argument/value pairs), graphically (plot with the Vertical Line Test: a curve is a function's graph iff every vertical line meets it at exactly one point), or analytically (a formula, whose natural domain is wherever that formula is actually defined -- found by excluding zero denominators, requiring even-root radicands $\ge0$, etc., often via a sign-chart over intervals cut out by the critical points). Functions may also be piecewise (different formula on different sub-intervals).

Named elementary functions: identity ($f(x)=x$), constant (and the zero function as its special case), modulus $|x|$, signum $x/|x|$ (with $0\mapsto0$), floor $\lfloor x\rfloor$ (always rounds down, even for negatives) and ceiling $\lceil x\rceil$ (always rounds up) -- the last two are "step functions". …