Let f : R→R be a function defined by f(x)


Let $f: R \rightarrow R$ be a function defined by $f(x)=\frac{x^{2}-8}{x^{2}+2}$. Then, $f$ is

(a) one-one but not onto
(b) one-one and onto
(c) onto but not one-one
(d) neither one-one nor onto


Let x and y be two elements in the domain (R), such that


$\Rightarrow \frac{x^{2}-8}{x^{2}+2}=\frac{y^{2}-8}{y^{2}+2}$


$\Rightarrow x^{2} y^{2}+2 x^{2}-8 y^{2}-16=x^{2} y^{2}-8 x^{2}+2 y^{2}-16$

$\Rightarrow 10 x^{2}=10 y^{2}$

$\Rightarrow x^{2}=y^{2}$


$\Rightarrow x=\pm y$

So, f is not one-one.



and $f(1)=\frac{(1)^{2}-8}{(1)^{2}+2}=\frac{1-8}{1+2}=\frac{-7}{3}$

$\Rightarrow f(-1)=f(1)=\frac{-7}{3}$

$\Rightarrow f$ is not onto. The correct answer is $(d)$.

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