Linear Algebra, 5th Edition

Appendix B Functions

If A and B are sets, then a function f from A to B, written $f:A\to B$, is a rule that associates to each element x in A a unique element denoted f(x) in B. The element f(x) is called the image of x (under f), and x is called a preimage of f(x) (under f). If $f:A\to B$, then A is called the domain of f, B is called the codomain of f, and the set $\{f(x):x\in A\}$ is called the range of f. Note that the range of f is a subset of B. If $S\subseteq A$, we denote by f(S) the set $\{f(x):x\in S\}$ of all images of elements of S. Likewise, if $T\subseteq B$, we denote by $f^{-1}(T)$ the set $\{x\in A:f(x)\in T\}$ of all preimages of elements in T. Finally, two functions $f:A\to B$ and $g:A\to B$ are equal, written $f=g$, if $f(x)=g(x)$ for all $x\in A$.

Example 1

Suppose that $A=[-10,10]$. Let $f:A$