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Onto & Into Functions

☼ Onto & Into Functions : Let f : A $\large \rightarrow$ B. We say that f is onto, if every element in B has at least one pre-image in A.

Clearly, f is onto $\large \Leftrightarrow$ $R_{f}$ = f (A) = { f (x) : x $\in$ A } = B.

On the other hand, if there exists even a single element in B having no pre-image in A, we say that f is into.

The function f has inverse if and only if ( or iff or $\large \Leftrightarrow$ ) f is one – one and onto.