# Explanation: Some Remarks on Functions

(related to Chapter: Functions (Maps))

We want to visualize the introduced concepts on an example. The function $f:A\mapsto B$ maps the elements of the set $A$ into the elements of the set $B$. It is left-total, since all elements $x_1,x_2,x_3,x_4 of$A$are mapped. It is right-unique, since every$x\in A$has only one value$f(x)\in B$. As the figure shows, it is not necessary for a function to be right-total, since not all elements of$B$have to be values of some elements of$A$: In the example,$y_2,y_4$and$y_5$are not contained in the image$f[A]$. In general, the image$f[A]$is a proper subset of but not equal$B$(in formulae$f[A]\subset B$but$f[A]\neq B$). Moreover, a function is in general, not left-unique, i.e. the values in$B$have not to be different for different elements in$A$. For instance,$f(x_3)=f(x_4)$. For this reason, the fiber$f^{-1}(y_r)$contains two elements$x_3,x_4\in A.$. Caution: The fiber$f^{-1}$is not a function, it is only a notation for a special set. There are other mathematical objects with the same notation. Usually, inverse functions and the function $$x\mapsto \frac1{f(x)}$$ which we will learn about later, are also denoted by$f^{-1}\$. You must be careful not to confuse these concepts.

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