Question

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Answer 1

Solution :-

Given that :-

$f(x) = log\left(\dfrac{1+x}{1-x}\right)$

-1 < x < 1 .

Now by applying the log property.

$f(x) = log(1+x) - log(1-x)$

Now we will Substitute

$x \: by \: \dfrac{3x + x^3}{1 + 3x^2}$

$\small{\rightarrow f\left(\dfrac{3x + x^3}{1 + 3x^2}\right) = log\left(1+\dfrac{3x + x^3}{1 + 3x^2}\right) - log\left(1-\dfrac{3x + x^3}{1 + 3x^2}\right)}$

$\small{\rightarrow f\left(\dfrac{3x + x^3}{1 + 3x^2}\right) = log\left(\dfrac{x^3 + 3x^2 + 3x + 1}{1 + 3x^2}\right) - log\left(\dfrac{-(x^3 - 3x^2 + 3x - 1)}{1 + 3x^2}\right)}$

$\small{\rightarrow f\left(\dfrac{3x + x^3}{1 + 3x^2}\right) = log\left(\dfrac{(x+1)^3}{1 + 3x^2}\right) - log\left(\dfrac{(-)(x-1)^3}{1 + 3x^2}\right)}$

$\small{\rightarrow f\left(\dfrac{3x + x^3}{1 + 3x^2}\right) = log[(x+1)^3] - log[(1+3x^2)] - log[(-)(x-1)^3] + log[(1+3x^2)] }$

$\rightarrow f\left(\dfrac{3x + x^3}{1 + 3x^2}\right) = 3log[(x+1)] - 2log[(x-1)] - log[(1-x)]$

Now

$x \: by \: \dfrac{2x}{1 + x^2}$

$\small{\rightarrow f\left(\dfrac{2x}{1 + x^2}\right) = log\left(1+\dfrac{2x }{1 + x^2}\right) - log\left(1-\dfrac{2x }{1 + x^2}\right)}$

$\small{\rightarrow f\left(\dfrac{2x }{1 + x^2}\right) = log\left(\dfrac{ x^2 + 2x + 1}{1 + 3x^2}\right) - log\left(\dfrac{x^2 - 3x +1}{1 + 3x^2}\right)}$

$\small{\rightarrow f\left(\dfrac{2x }{1 + x^2}\right) = log\left(\dfrac{(x+1)^2}{1 + 3x^2}\right) - log\left(\dfrac{(x-1)^2}{1 + 3x^2}\right)}$

$\small{\rightarrow f\left(\dfrac{2x}{1 + x^2}\right) = log[(x+1)^2] - log[(1+x^2)] - log[(x-1)^2] + log[(1+x^2)] }$

$\rightarrow f\left(\dfrac{2x}{1 + x^2}\right) = 2log[(x+1)] - 2log[(x-1)]$

Now

$f\left(\dfrac{3x + x^3}{1 + 3x^2}\right) - f\left(\dfrac{2x}{1 + x^2}\right)$

$\small{= 3log[(x+1)] - 2log[(x-1)] - log[(1-x)] -( 2log[(x+1)] - 2log[(x-1)] )}$

$\small{=3log[(x+1)] - 2log[(x-1)] - log[(1-x)] - 2log[(x+1)] + 2log[(x-1)]}$

$= log[(x+1)] - log[(1-x)]$

$= log\left(\dfrac{1+x}{1-x}\right)$

$= f(x)$

So Answer is option (d)

Answer 2

Step-by-step explanation:

[refer the attachments for the Explanation ]

ANS : option d ] (f(x)) is correct................

formulae used :

(a + b)³ = a³ + 3a²b + 3ab² + b³

(a - b)³ = a³ - 3a²b + 3ab² - b³

(a + b)² = a² + 2ab + b²

(a - b)² = a² - 2ab + b²

log rule :

log (a/b) = log a - log b