Functions Question 551
Question: If $ f(x)=\frac{ax+d}{cx+b} $ and $ f[f(x)]=x $ for all x, then:
Options:
A) $ a=b $
B) $ c=d $
C) $ a+b=0 $
D) $ c+d=0 $
Show Answer
Answer:
Correct Answer: C
Solution:
[c] $ f(x)=\frac{ax+d}{cx+b} $ $ f(f(x))=\frac{a( \frac{ax+d}{cx+b} )+d}{c( \frac{ax+d}{cx+b} )+b}=\frac{a^{2}x+ad+cdx+bd}{cax+cd+bcx+b^{2}} $ $ f(f(x))=x\Rightarrow \frac{a^{2}x+ad+cdx+bd}{cax+cd+bcx+b^{2}}=x $
$ \Rightarrow c(a+b)x^{2}-(a^{2}-b^{2})x-(a+b)d=0 $
$ \Rightarrow (a+b)(cx^{2}-(a-b)x-d)=0\Rightarrow a+b=0 $ As $ cx^{2}-(a-b)x-d\ne 0 $ for all x
BETA
