The function , defined by is : [2025]

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Solution

Q.
The function $f : (- \infty, \infty) \to (- \infty, 1)$ , defined by $f (x) = \frac{2^{x} - 2^{- x}}{2^{x} + 2^{- x}}$ is : [2025]

1 Onto but not one-one

2 Both one-one and onto

3 One-one but not onto

4 Neither one-one nor onto

Ans.
(3)

Given function is $f (x) = \frac{2^{x} - 2^{- x}}{2^{x} + 2^{- x}}$

$f (x) = \frac{2^{x} - \frac{1}{2^{x}}}{2^{x} + \frac{1}{2^{x}}} = \frac{2^{2 x} - 1}{2^{2 x} + 1} = 1 - \frac{2}{2^{2 x} + 1}$

$\Rightarrow f' (x) = \frac{2}{{(2^{2 x} + 1)}^{2}} \times 2 \times 2^{2 x} \times \log_{e} (2) > 0$

$\Rightarrow$ f(x) is always increasing.

Hence it is one-one.

Since $f (- \infty) = - 1$ and $f (\infty) = 1 \Rightarrow f (x) \in (- 1, 1) \neq (- \infty, 1)$

Thus, the function f(x) is one-one but not onto.

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