Question

Let A = R - {3}, B = R - {1}. Let f : A $\to$ B be defined by f(x) = $\frac{x-2}{x-3}$ Then :

• A. f is bijective
• B. f is one-one but not onto
• C. f is onto but not one-one
• D. none of these

Answer 1

Answer: (A)

Let $f(x)=\frac{x-2}{x-3}$
To show : f (x) is one - one
Let f (x) = f (y)
$\Rightarrow \frac{x-2}{x-3}=\frac{y-2}{y-3}$ To show : x = y
$\Rightarrow xy-3x-2y+6=xy-2x-3y+6$
$\Rightarrow -3x-2y=-2x-3y$
$\Rightarrow y=x$ (proved)
Hence, f (x) is one - one
To show : f (x) is an onto function
Let y $\in$ Co-domain (B) (To show : There exist x $\in$ Domain such that f(x) = y) consider f (x) = y
$\Rightarrow \frac{x-2}{x-3}=y\Rightarrow x-2=xy-3y$
$\Rightarrow x(1-y)=2-3y\Rightarrow x=\frac{2-3y}{1-y}\in A$
Clearly, for all y $\in$ B we have
$x=\frac{2-3y}{1-y}\in A$
$\therefore$ f(x) is an onto function
Thus f(x) is one - one and onto function.
$\Rightarrow$ f(x) is Bijective function.