The function f:(- , ) (- , 1), defined by f(x)= 2^ x -2^ -x 2^ x … - Vidyadip

MCQ

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

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

✓

Answer

Correct option: A.

One-one but not onto

(A)
Sol. $f(x)=\frac{2^{2 x}-1}{2^{2 x}+1}$
$=1-\frac{2}{2^{2 \mathrm{x}}+1}$
$f^{\prime}(x)=\frac{2}{\left(2^{2 x}+1\right)^{2}} \cdot 2 \cdot 2^{2 x} \cdot \ln 2$ i.e always $+v e$
so $f(x)$ is $\uparrow$ function
$\therefore \mathrm{f}(-\infty)=-1$
$f(\infty)=1$
$\therefore \mathrm{f}(\mathrm{x}) \in(-1,1) \neq$ co-domain
so function is one-one but not onto

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