State whether the function is one-one, onto or bijective. Justify…

Question

State whether the function is one-one, onto or bijective. Justify your answer. f: R $\rightarrow$ R defined by f(x) = 1+ x²

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Answer

Let x₁, x₂ $\in$ R
If f(x₁) = f(x₂)
$1 + x_1^2 = 1 + x_1^2$
$x_1^2 = x_1^2$
${x_1} = \pm {x_2}$
Hence not one - one
y = 1 + x²
$x = \pm \left( {\sqrt {1 - y} } \right)$
$f\left( {\sqrt {1 - y} } \right) = 1 + (1 - y) = 2 - y \ne y$
Therefore, f is not onto.

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