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Question
\[f : R \to R\] is defined by
\[f\left( x \right) = \frac{e^{x^2} - e^{- x^2}}{e^{x^2 + e^{- x^2}}} is\]
Options
one-one but not onto
many-one but onto
one-one and onto
neither one-one nor onto
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Solution
(d) neither one-one nor onto
\[We have, \]
\[f\left( x \right) = \frac{e^{x^2} - e^{- x^2}}{e^{x^2 + e^{- x^2}}}\]
\[\text{Here}, - 2, 2 \in R\]
\[Now, 2 \neq - 2\]
\[\text{But}, f\left( 2 \right) = f\left( - 2 \right)\]
\[\text{Therefore, function is not one - one} . \]
\[\text{And}, \]
\[\text{The minimum value of the function is 0 and maximum value is} 1\]
\[\text{That is range of the function is} \left[ 0, 1 \right] \text{but the co - domain of the function is given } R . \]
\[\text{Therefore, function is not onto} . \]
\[ \therefore \text{function is neither one - one nor onto} . \]
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