The function f:(-∞, ∞) rightarrow(-∞, 1) , defined by f(x)=frac 2 x -2 -x 2 x +2 -x is :

Mathematics · Functions

JEE Main 2025 — 24 January, Evening Shift — Question 2

The function $f:(-\infty, \infty) \rightarrow(-\infty, 1)$ , defined by $f(x)=\frac{2^{x}-2^{-x}}{2^{x}+2^{-x}}$ is :

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

=1-\frac{2}{2^{2 x}+1}

$f^{\prime}(x)=\frac{2}{\left(2^{2 x}+1\right)^{2}} \cdot 2 \cdot 2^{2 x} \cdot \ln 2$ i.e always $+v e$

so $\mathrm{f}(\mathrm{x})$ is $\uparrow$ function

$\therefore \mathrm{f}(-\infty)=-1$

$\mathrm{f}(\infty)=1$

$\therefore \mathrm{f}(\mathrm{x}) \in(-1,1) \neq$ co-domain

so function is one-one but not onto

Answer key and solution verified before publishing.

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