MHT CET · Maths · Functions
Let \(f(x)=\frac{a x+b}{c x+d}\). Then, \(f o f(x)=x\) provided
that
- A \(d=-a\)
- B \(d=a\)
- C \(a=b=c=d=1\)
- D \(a=b=1\)
Answer & Solution
Correct Answer
(A) \(d=-a\)
Step-by-step Solution
Detailed explanation
Given,
and \(f o f(x)=x\)
\(
\Rightarrow f\left(\frac{a x+b}{c x+d}\right)=x
\)
\(
\begin{array}{l}
\Rightarrow \frac{a\left(\frac{a x+b}{c x+d}\right)+b}{c\left(\frac{a x+b}{c x+d}\right)+d}=x \\
\Rightarrow \frac{x\left(a^{2}+b c\right)+a b+b d}{x(a c+c d)+b c+d^{2}}=x \\
\Rightarrow d=-a
\end{array}
\)
and \(f o f(x)=x\)
\(
\Rightarrow f\left(\frac{a x+b}{c x+d}\right)=x
\)
\(
\begin{array}{l}
\Rightarrow \frac{a\left(\frac{a x+b}{c x+d}\right)+b}{c\left(\frac{a x+b}{c x+d}\right)+d}=x \\
\Rightarrow \frac{x\left(a^{2}+b c\right)+a b+b d}{x(a c+c d)+b c+d^{2}}=x \\
\Rightarrow d=-a
\end{array}
\)
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