Question

The function defined by $f\left(x\right)=\frac{ax+b}{cx+d}$, where a,b,c,d are non zero real numbers, f(42) = 42, f(50) = 50, f(f(x)) = x for all values except - $\frac{d}{c}$ then the numbers which are possible in range of f(x) is (are)

• A. 40
• B. 46
• C. 48
• D. 52

Answer 1

Answer: (A), (C), (D)

The correct options are
A 40
C 48
D 52

Domain of $f\left(x\right)=R-\{-\frac{d}{c}\}$
Range of $f\left(x\right)=R-\left\{\frac{a}{c}\right\}$
Since f(x) is inverse of itself
Domain = Range
$d=-af\left(x\right)=\frac{ax+b}{cx-a}$
$f\left(42\right)=42\&f\left(50\right)=50\Rightarrow \frac{ax+b}{cx-a}=x\text{has two roots 42 \& 50.}$
$\Rightarrow ax+b=c{x}^2-ax\Rightarrow c{x}^2-2ax-b=0\text{has two roots 42 \& 50.}$
$\frac{2a}{c}=(42+50)\Rightarrow \frac{a}{c}=46$
$\frac{a}{c}$ is not possible in range.