Question

Let the function $f:R-\{-b\}\to R-\left\{1\right\}$ be defined by $f\left(x\right)=\frac{x+a}{x+b},a\ne b.$ Then

• A. 𝑓 is one–one but not onto function
• B. 𝑓 is onto but not one–one function
• C. 𝑓 is bijective function
• D. 𝑓 is neither one–one nor onto function

Answer 1

The correct option is C 𝑓 is bijective function

For one-one :
Let $x,y\in R-\{-b\}$ such that $f\left(x\right)=f\left(y\right)$
$\Rightarrow \frac{x+a}{x+b}=\frac{y+a}{y+b}$
$\Rightarrow xy+ay+bx+ab=xy+xa+yb+ab$
$\Rightarrow x(b-a)=y(b-a)\Rightarrow x=y(\because a\ne b)$
$\therefore f$ is one-one function.

For onto :
Let $y\in R$ such that $f\left(x\right)=y$
$\Rightarrow \frac{x+a}{x+b}=y$
$\Rightarrow x+a=xy+yb$
$\Rightarrow x=\frac{a-by}{y-1}$
$\Rightarrow y\in R-\left\{1\right\}$
$\therefore f$ is onto function.

Hence, $f$ is one-one and onto function, it is bijective function.