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(C) The given function is $f(x) = \frac{ax + b}{cx + d}$.For the function to be a constant function,its derivative with respect to $x$ must be zero,or

Vedclass · Accepted Answer

$ad = bc$

Vedclass · Answer

(C) The given function is $f(x) = \frac{ax + b}{cx + d}$.For the function to be a constant function,its derivative with respect to $x$ must be zero,or the numerator must be a constant multiple of the denominator.Let $f(x) = k$ (a constant).Then $\frac{ax + b}{cx + d} = k$.$ax + b = k(cx + d) = (kc)x + kd$.Comparing the coefficients of $x$ and the constant terms,we get $a = kc$ and $b = kd$.This implies $\frac{a}{c} = k$ and $\frac{b}{d} = k$ (assuming $c, d 
eq 0$).Therefore,$\frac{a}{c} = \frac{b}{d}$,which gives $ad = bc$.Alternatively,if we take option $(c)$,i.e.,$ad = bc$,then $ad - bc = 0$.If $ad = bc$,then $\frac{a}{c} = \frac{b}{d} = k$ (for $c, d 
eq 0$).Substituting $a = kc$ and $b = kd$ into the function:$f(x) = \frac{(kc)x + kd}{cx + d} = \frac{k(cx + d)}{cx + d} = k$.Since $f(x) = k$,the function is constant.

The function $f: C \rightarrow C$ defined by $f(x) = \frac{ax + b}{cx + d}$ for $x \in C$,where $ad - bc \neq 0$,reduces to a constant function if:

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