Types of Functions

One-to-one and onto. For $f:A\to B$:

• $f$ is one-to-one (injective) if distinct domain elements always have distinct images: $x\ne y\Rightarrow f(x)\ne f(y)$ -- equivalently (and usually easier to use in a proof), $f(x)=f(y)\Rightarrow x=y$.
• $f$ is onto (surjective) if every element of $B$ has at least one pre-image: for each $b\in B$ there is some $a\in A$ with $f(a)=b$ -- equivalently, the range of $f$ equals the whole co-domain $B$.
• $f$ is a bijection if it is both one-to-one and onto. A function that fails to be onto is called an into function (its range is a proper subset of the co-domain).

Every identity function is a bijection; a constant function is onto only if the co-domain is a single element.

Finite-set cardinality rules. For $A,B$ finite with $n(A)=m,\ n(B)=n$:

(i) no one-to-one function $A\to B$ exists if $m>n$; (ii) a one-to-one function existing forces $m\le n$; (iii) no onto function exists if $m<n$; (iv) an onto function existing forces $m\ge n$; (v) a bijection exists iff $m=n$; (vi) no bijection exists iff $m\ne n$.

Both the domain and the co-domain matter for ontoness (changing either can flip the answer); only the domain and rule matter for one-to-oneness (the co-domain is irrelevant to injectivity). E.g. $f:N\to N,\ f(n)=n+2$ is one-to-one but not onto ($1$ has no pre-image); enlarging the domain to $N\cup\{-1,0\}$ makes the same rule onto as well, because now every $m\in N$ has pre-image $m-2$.

Horizontal Line Test. For a function given as a curve: it is one-to-one iff every horizontal line through a range value meets the curve at exactly one point; it is onto (onto its stated co-domain) iff every horizontal line through a co-domain value meets the curve at least once. So $f(x)=2x$ on $R\to R$ is a bijection; $f(x)=x^2$ on $R\to R$ is neither injective nor surjective; $f(x)=\sqrt x$ on $[0,\infty)\to R$ is injective but not surjective, while the same rule $[0,\infty)\to[0,\infty)$ is a bijection. …