Question

If f : R - {±1} → R is defined by f(x) = $log |\frac{1 + x}{1 - x}|$, then show that $f [ \frac{2x}{1 + x^{2}} ]$ = 2 f(x).

Answer 1

Answer:

f(2x/1 + x²) = 2 f(x)

Step-by-step explanation:

$Given: f(x)=log(\frac{1+x}{1-x})$

∴$f(\frac{2x}{1+x^2}):$

$=log(\frac{1+\frac{2x}{1+x^2}}{1-\frac{2x}{1+x^2}})$

$=log(\frac{1+x^2 + 2x}{1+x^2-2x})$

$=log(\frac{1+x}{1-x})^2$

$=2log(\frac{1+x}{1-x})$

$=\boxed{2f(x)}$

Hope it helps!