Q79 / 155 · WBJEE 2024 · Mathematics

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function defined by $f(x)=\frac{e^{|x|}-e^{-x}}{e^x+e^{-x}}$, then

  • A
    f is both one-one and onto
  • B
    f is one-one but not onto
  • C
    f is onto but not one-one
  • Correct answer
    D
    f is neither one-one nor onto

Solution

Hint: $f: \mathbb{R} \rightarrow \mathbb{R}, f(x)=\frac{e^{|x|}-e^{-x}}{e^x+e^{-x}}$ Case I: If $x \geq 0$ $f(x)=\frac{e^x-e^{-x}}{e^x+e^{-x}}=1-\frac{2 e^{-x}}{e^x+e^{-x}}=1-\frac{2}{e^{2 x}+1}\gt-1$ Case II: $x \lt 0$ $f(x)=\frac{e^{-x}-e^{-x}}{e^x+e^{-x}}=0$

Hence, the correct answer is option: 'D'.

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