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Let $f(x)=\frac{a x+b}{c x+d}$. Then, $f o f(x)=x$ provided
that
Options:
that
Solution:
2627 Upvotes
Verified Answer
The correct answer is:
$d=-a$
Given,
and $\quad f o f(x)=x$
$$
\Rightarrow \quad f\left(\frac{a x+b}{c x+d}\right)=x
$$
$$
\begin{array}{l}
\Rightarrow \quad \frac{a\left(\frac{a x+b}{c x+d}\right)+b}{c\left(\frac{a x+b}{c x+d}\right)+d}=x \\
\Rightarrow \quad \frac{x\left(a^{2}+b c\right)+a b+b d}{x(a c+c d)+b c+d^{2}}=x \\
\Rightarrow \quad d=-a
\end{array}
$$
and $\quad f o f(x)=x$
$$
\Rightarrow \quad f\left(\frac{a x+b}{c x+d}\right)=x
$$
$$
\begin{array}{l}
\Rightarrow \quad \frac{a\left(\frac{a x+b}{c x+d}\right)+b}{c\left(\frac{a x+b}{c x+d}\right)+d}=x \\
\Rightarrow \quad \frac{x\left(a^{2}+b c\right)+a b+b d}{x(a c+c d)+b c+d^{2}}=x \\
\Rightarrow \quad d=-a
\end{array}
$$
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