Question

The function $f\left(x\right)$ defined on the real numbers has the property that $f\left(f\right(x\left)\right).(1+f(x\left)\right)=-f\left(x\right)$ for all $x$ in the domain of $f$. If the number $3$ is in the domain and range of $f$, compute the value of $f\left(3\right)$.

• A. $-\frac{3}{4}$
• B. $-\frac{3}{2}$
• C. $\frac{2}{3}$
• D. $-\frac{2}{3}$

Answer 1

The correct option is A $-\frac{3}{4}$

$f\left(f\right(x\left)\right)[1+f(x\left)\right]=-f\left(x\right)$

$f\left(f\right(x\left)\right)=\frac{-f\left(x\right)}{1+f\left(x\right)}\Rightarrow f\left(x\right)=\frac{-x}{1+x}$

$\therefore f\left(3\right)=\frac{-3}{1+3}=\frac{-3}{4}$