Kompozicija funkcija

Ako je

$\displaystyle f(x)=\log{\frac{1+x}{1-x}}\quad\textrm{i}\quad g(x)=\frac{3x+x^3}{3x^2+1},$

dokažite da je $\left(f\circ g\right)(x)=3f(x)$ .

Rješenje. Prema [M1, definicija 1.9] slijedi

$\displaystyle \left(f\circ g\right)(x)$	$\displaystyle =f\left(g(x)\right)=f\left(\frac{3x+x^3}{3x^2+1}\right)=\log{\frac{1+\displaystyle\frac{3x+x^3}{3x^2+1}}{1-\displaystyle\frac{3x+x^3}{3x^2+1}}}$
	$\displaystyle =\log{\frac{\displaystyle\frac{x^3+3x^2+3x+1}{3x^2+1}}{\displaystyle\frac{-x^3+3x^2-3x+1}{3x^2+1}}} =\log{\frac{x^3+3x^2+3x+1}{-x^3+3x^2-3x+1}}$
	$\displaystyle =\log{\frac{(x+1)^3}{(1-x)^3}}=3\log{\frac{x+1}{1-x}}=3f(x).$