Let be a function defined by . Then

JEE MainMathematicsModerateFunctions

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

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$f(x) = \frac{e^{|x|} - e^{- x}}{e^{x} + e^{- x}} = \{\begin{matrix}
\frac{e^{x} - e^{- x}}{e^{x} + e^{- x}} &; & x \geq 0 \\
0 &; & x < 0
\end{matrix}$

\Rightarrow f(x)

cannot take negative values.

\Rightarrow f(x)

is many-one into function.

(\therefore forx > 0)e^{x} > e^{- x} \Rightarrow e^{x} - e^{- x} > 0\therefore\frac{e^{x} - e^{- x}}{e^{x} + e^{- x}} > 0

Source: Grass-G / M-17 Mathematics Paper 26 Nov Second.docx

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Let f:R→Rf:R \rightarrow Rf:R→R be a function defined byf(x)=e∣x∣−e−xex+e−xf(x) = \frac{e^{|x|} - e^{- x}}{e^{x} + e^{- x}}f(x)=ex+e−xe∣x∣−e−x​. Then

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