Question

Let $f:R\to R$ defined by $f\left(x\right)=\frac{x^2-8}{x^2+2},$ then $f\left(x\right)$ is

• A. onto but not one-one function
• B. neither one-one nor onto function
• C. one-one but not onto function
• D. bijective function

Answer 1

The correct option is B neither one-one nor onto function

We have,
$f\left(x\right)=\frac{x^2-8}{x^2+2}$
$\Rightarrow f(-x)=\frac{x^2-8}{x^2+2}=f\left(x\right)$
$\therefore$ It is not a one-one function.

Now, $f\left(x\right)=\frac{x^2-8}{x^2+2}=1-\frac{10}{x^2+2}<1$
It is also not onto function because pre-images of $1,2,3,\dots$ do not exist.

Hence, $f\left(x\right)$ is neither one-one nor onto function.