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Q.
Which of the following statement(s) is(are) correct?
Relations and Functions - Part 2
Solution:
(A) Let $f: N \rightarrow N$ such that $f(x)=2 x$. Clearly $f$ is one-one but not onto.
Note: If $f$ is a one-one mapping from set $A$ to $A$, then $f$ is onto only if $A$ is finite set.
(B) $ f : R \rightarrow R$ such that $f ( x )= x ^3- x ^2-4 x +4$. Clearly $f (-2)= f (2)= f (1)=0$.
Hence $f$ is many one but since it is an odd degree polynomial therefore its range is $R$ hence it is onto.
Note: If $f$ is a onto mapping from set $A$ to $A$ then $f$ is one-one only if $A$ is finite set.
(C) Suppose $f$ is not one-one then there are atleast two real numbers $x _1, x _2 \in R , x _1 \neq x _2$ such that $f \left( x _1\right)= f \left( x _2\right)$
$\therefore g \left( f \left( x _1\right)\right)= g \left( f \left( x _2\right)\right)$
i.e. gof is not one-one which is a contradiction to the given hypothesis that gof is one-one.
Hence $f$ must be one-one.
(D) Clearly, total number of functions from A to $B =2 \times 2 \times 2=8$.