QJEEM25Ma80da: JEE Main 2025 Mathematics - Class 11 JEEM25Ma80da Solution: The function , defined by is :

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QJEEM25Ma80da: JEE Main 2025 Mathematics - Class 11 JEEM25Ma80da Solution: The function , defined by is :

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

Multiple Choice Options

A One-one but not onto
B Onto but not one-one
C Both one-one and onto
D Neither one-one nor onto

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Accepted Answer

$\begin{aligned} f (x) = \frac{2^{2 x} - 1}{2^{2 x} + 1} \\ = 1 - \frac{2}{2^{2 x} + 1} \\ f^{'} (x) = \frac{2}{{(2^{2 x} + 1)}^{2}} \cdot 2 \cdot 2^{2 x} \cdot \ln 2 i.e always + ve \end{aligned}$

so $f(x)$ is $\uparrow$ function

$\begin{aligned} ∴ f (- \infty) = - 1 \\ f (\infty) = 1 \\ ∴ f (x) \in (- 1, 1) \neq co-domain \end{aligned}$

so function is one-one but not onto

QJEEM25Ma80da: JEE Main 2025 Mathematics - Class 11 JEEM25Ma80da Solution: The function , defined by is :

Multiple Choice Options

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