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For a suitably chosen real constant $a$, let a function, $f: R-\{-a\} \rightarrow R$ be defined by $f(x)=\frac{a-x}{a+x} .$ Further suppose that for any real number $x \neq- a$ and $f( x ) \neq- a ,( fof )( x )= x .$ Then $f\left(-\frac{1}{2}\right)$ is equal to
$\frac{1}{3}$
$3$
$-3$
$-\frac{1}{3}$
Solution
$f(x)=\frac{a-x}{a+x} \quad x \in R-\{-a\} \rightarrow R$
$f(f(x))=\frac{a-f(x)}{a+f(x)}=\frac{a-\left(\frac{a-x}{a+x}\right)}{a+\left(\frac{a-x}{a+x}\right)}$
$f(f(x))=\frac{\left(a^{2}-a\right)+x(a+1)}{\left(a^{2}+a\right)+x(a-1)}=x$
$\Rightarrow\left(a^{2}-a\right)+x(a+1)=\left(a^{2}+a\right) x+x^{2}(a-1)$
$\Rightarrow a(a-1)+x\left(1-a^{2}\right)-x^{2}(a-1)=0$
$\Rightarrow a=1$
$f(x)=\frac{1-x}{1+x}$
$f\left(\frac{-1}{2}\right)=\frac{1+\frac{1}{2}}{1-\frac{1}{2}}=3$
