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Question
The function
\[f : R \to R\] defined by\[f\left( x \right) = \left( x - 1 \right) \left( x - 2 \right) \left( x - 3 \right)\]
(a) one-one but not onto
(b) onto but not one-one
(c) both one and onto
(d) neither one-one nor onto
Options
one-one but not onto
onto but not one-one
both one and onto
neither one-one nor onto
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Solution
\[f\left( 2 \right) = \left( 2 - 1 \right)\left( 2 - 2 \right)\left( 2 - 3 \right) = 0\]
\[f\left( 3 \right) = \left( 3 - 1 \right)\left( 3 - 2 \right)\left( 3 - 3 \right) = 0\]
\[\Rightarrow f \left( 1 \right) = f\left( 2 \right) = f\left( 3 \right) = 0\]
Let y be an element in the co domain R, such that
\[ \Rightarrow y = \left( x - 1 \right)\left( x - 2 \right)\left( x - 3 \right)\]
\[\text{Sincey} \in R \text{ and }x \in R, \text{f is onto}.\]
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