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Question
Let
Options
one-one but not onto
one-one and onto
onto but not one-one
neither one-one nor onto
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Solution
Injectivity:
Let x and y be two elements in the domain (R), such that
\[f\left( x \right) = f\left( y \right)\]
\[ \Rightarrow \frac{x^2 - 8}{x^2 + 2} = \frac{y^2 - 8}{y^2 + 2}\]
\[ \Rightarrow \left( x^2 - 8 \right)\left( y^2 + 2 \right) = \left( x^2 + 2 \right)\left( y^2 - 8 \right)\]
\[ \Rightarrow x^2 y^2 + 2 x^2 - 8 y^2 - 16 = x^2 y^2 - 8 x^2 + 2 y^2 - 16\]
\[ \Rightarrow 10 x^2 = 10 y^2 \]
\[ \Rightarrow x^2 = y^2 \]
\[ \Rightarrow x = \pm y \]
So, f is not one-one .
Surjectivity:
\[ \text{ and} f\left( 1 \right) = \frac{\left( 1 \right)^2 - 8}{\left( 1 \right)^2 + 2} = \frac{1 - 8}{1 + 2} = \frac{- 7}{3}\]
The correct answer is (d).
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