First slide

Functions (XII)

Question

Let $f : R \to R$ be a function defined by $f (x) = \frac{e^{| x |} - e^{- x}}{e^{x} + e^{- x}}$ . Then

Moderate

A

$f$ is both one-one and onto
B

$f$ is one-one but not onto
C

$f$ is onto but not one-one
D

$f$ is neither one-one nor onto

Solution

$f$ is not one-one as f (0) = 0 and f (-1) = 0.
$f$ is also not onto as for $y = 1$ there is no $x \in R$ such that f (x) = 1.
If there is such a $x \in R$ then $e^{| x |} - e^{- x} = e^{x} + e^{- x},$ clearly $x \neq 0 .$
For x > 0, this equation gives $- e^{- x} = e^{- x}$ which is not possible. For x < 0,
the above equation gives $e^{x} = - e^{- x}$ which is also not possible .

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