Classify the following functions as injection, surjection or bije…

Question

Classify the following functions as injection, surjection or bijection:$f : R \rightarrow R,$ defined by $f(x) = 1 + x^2$

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Answer

$f : R → R,$ defined by $f(x) = 1 + x^2$
Injection test: Let $\text{x, y}\in\text{R,}$ such that,
$f(x) = f(y)$
$\Rightarrow 1 + x^2 = 1 + y^2$
$\Rightarrow x^2 - y^2 = 0$
$\Rightarrow (x - y)(x + y) = 0$
either $x = y$ or $x = -y$ or $\text{x}\neq\text{y}$
Therefore, f is not one-one.
Surjection: Let $\text{y}\in\text{R}$ be arbitrary, then
$f(x) = y$
$\Rightarrow 1 + x^2 = y$
$\Rightarrow x^2 + 1 - y = 0$
$\therefore\ \text{x}\pm\sqrt{\text{y}-1}\notin\text{R}$ or $y < 1$
$\therefore f $ is not onto.

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