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Functions — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsFunctions2 Marks
MCQ
The inverse of the function $f (x)=\frac{ e ^x- e ^{-x}}{ e ^x+ e ^{-x}}+2$ is
  • A
    $\log _e\left(\frac{x-2}{x-1}\right)^{\frac{1}{2}}$
  • ✓
    $\log _e\left(\frac{x-1}{3-x}\right)^{\frac{1}{2}}$
  • C
    $\log _e\left(\frac{x}{2-x}\right)^{\frac{1}{2}}$
  • D
    $\log _e\left(\frac{x-1}{x+1}\right)^{-2}$
✓

Answer

Correct option: B.
$\log _e\left(\frac{x-1}{3-x}\right)^{\frac{1}{2}}$
(B)
Let $y= f (x)=\frac{ e ^x- e ^{-x}}{ e ^x+ e ^{-x}}+2$
$\therefore \quad y-2=\frac{ e ^{2 x}-1}{ e ^{2 x}+1}$
$\Rightarrow(y-2) e ^{2 x}+y-2= e ^{2 x}-1$
$\Rightarrow e ^{2 x}=\frac{1-y}{y-3}=\frac{y-1}{3-y}$
$\Rightarrow 2 x=\log _{ e }\left(\frac{y-1}{3-y}\right)$
$\Rightarrow x=\frac{1}{2} \log _{ e }\left(\frac{y-1}{3-y}\right)$
$\Rightarrow f ^{-1}(y)=\frac{1}{2} \log _e\left(\frac{y-1}{3-y}\right)$
$\Rightarrow f ^{-1}(x)=\log _{ e }\left(\frac{x-1}{3-x}\right)^{\frac{1}{2}}$

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