Question

If $f\left(x\right)=\frac{ax+b}{cx+d}$ and $\left(fof\right)\left(x\right)=x$ then

• A. $a^2+bc=1$
• B. $d^2+bc=1$
• C. $a+d=0$
• D. none of these

Answer 1

Answer: (A), (B), (C)

The correct options are
A $a^2+bc=1$
B $d^2+bc=1$
C $a+d=0$

$f\left(x\right)=\frac{ax+b}{cx+d}$

$f\circ f\left(x\right)=\frac{a\left(\frac{ax+b}{cx+d}\right)+b}{c\left(\frac{ax+b}{cx+d}\right)+d}=x$

$\Rightarrow \frac{a(ax+b)+b(cx+d)}{c(ax+b)+d(cx+d)}=x$

$\Rightarrow \frac{x(a^2+bc)+(a+d)b}{cx(a+d)+(bc+d^2)}=\frac{x}{1}$

$\Rightarrow a^2+bc=1,a+d=0;bc+d^2=1$