Question

The function $f:\mathrm{ℂ}\to \mathrm{ℂ}$ defined by $f\left(x\right)=\frac{ax+b}{cx+d}$ for $x\in \mathrm{ℂ}$, where $bd\ne 0$, reduces to a constant function if

• A. $a=c$
• B. $b=d$
• C. $ad=bc$
• D. $ab=cd$

Answer 1

The correct option is C

$ad=bc$

Explanation for the correct option

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

If the function is a constant function, then it can be rewritten as, $f\left(x\right)=\frac{ax+b}{cx+d}=k$.

$\Rightarrow ax+b=kcx+kd$

As the derived equation is an identity, thus the coefficients of $x$ of both sides of the equation will be equal.

So, $a=kc$

$\Rightarrow k=\frac{a}{c}$.

The constant term of the identity will also be equal.

So, $b=kd$

$\Rightarrow k=\frac{b}{d}$

Hence, $k=\frac{a}{c}=\frac{b}{d}$.

$\Rightarrow ad=bc$.

Therefore, $f\left(x\right)=\frac{ax+b}{cx+d}$ can be reduced to a constant function if $ad=bc$.

Hence, $ad=bc$ and therefore the correct option is (C).