Question

Given f(x) = log `((1 + x)/(1 − x))` and g(x) = `(3x + x^3)/(1 + 3x^2)`, then fog(x) equals ______.

Options

• −f(x)

• 3f(x)

• [f(x)]³

• None of these

Answer 1

Given f(x) = log `((1 + x)/(1 − x))` and g(x) = `(3x + x^3)/(1 + 3x^2)`, then fog(x) equals 3f(x).

Explanation:

Since, g(x) = `3x + x^3/1 + 3x^2 = y,` (say)   ...[i]

∴ `f [g(x)] = (y) = log ((1 + y)/(1 − y))`

= `lo g {(1 + (3x + x^3)/(1 + 3x^2))/(1 − (3x + x^3)/(1 + 3x^2))}`

⇒ f (g(x)) = log`((1 + x)/(1 − x))^3`

= 3 lo g `((1 + x)/(1 − x)) = 3 f (x)`