WBJEE · Maths · Functions
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
- D f is neither one-one nor onto
Answer & Solution
Correct Answer
(D) f is neither one-one nor onto
Step-by-step Solution
Detailed explanation
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: \(x \lt 0\) \(f(x)=\frac{e^{-x}-e^{-x}}{e^x+e^{-x}}=0\)
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