The function f : (-∞, ∞)→(-∞, 1), defined by f(x)=2^x-2^(-x)/2^x+2^(-x) is:

Question

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

(1) One-one but not onto

(2) Onto but not one-one

(3) Both one-one and onto

(4) Neither one-one nor onto

Answer 1

Correct option is (1) One-one but not onto  

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

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

\(f^{\prime}(x)=\frac{2}{\left(2^{2 x}+1\right)^{2}} \cdot 2 \cdot 2^{2 x} \cdot \ell n 2\) 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