The inverse of f (x)= (2/3) (10x-10-x/10x+ 10-x) is

Question

The inverse of f (x)= (2/3) (10x-10-x/10x+ 10-x) is (A) (1/3) log10 (1+x/1-x) (B) (1/2) log10 (2+3x/2-3x) (C) (1/3) log10 (2+3x/2-3x) (D) (1/6) log10 (2-3x/2+3x)

Tardigrade Admin · Accepted Answer

If y = (2/3) (10x-10-x/10x+ 10-x), 102x = (3y+2/2-y) or x = (1/2) log10 (2+3y/2-3y) ∴ f-1 (x) = (1/2) log10 (2+3x/2-3x).

Q. The inverse of $f (x) = \frac{2}{3} \frac{1 0 ^{x} - 1 0 ^{- x}}{1 0 ^{x} + 1 0 ^{- x}}$ is

$\frac{1}{3} l o g_{10} \frac{1 + x}{1 - x}$

$\frac{1}{2} l o g_{10} \frac{2 + 3 x}{2 - 3 x}$

$\frac{1}{3} l o g_{10} \frac{2 + 3 x}{2 - 3 x}$

$\frac{1}{6} l o g_{10} \frac{2 - 3 x}{2 + 3 x}$

If $y = \frac{2}{3} \frac{1 0 ^{x} - 1 0 ^{- x}}{1 0 ^{x} + 1 0 ^{- x}}, 1 0^{2 x} = \frac{3 y + 2}{2 - y}$
or $x = \frac{1}{2} l o g_{10} \frac{2 + 3 y}{2 - 3 y} ∴ f^{- 1} (x) = \frac{1}{2} l o g_{10} \frac{2 + 3 x}{2 - 3 x} .$

Q. The inverse of f(x)=32​10x+10−x10x−10−x​ is

31​log10​1−x1+x​

21​log10​2−3x2+3x​

31​log10​2−3x2+3x​

61​log10​2+3x2−3x​