Question
Classify the following functions as injection, surjection or bijection:
f : R → R, defined by f(x) = 1 + x2
f : R → R, defined by f(x) = 1 + x2
Injection test: Let $\text{x, y}\in\text{R,}$ such that,
f(x) = f(y)
⇒ 1 + x2 = 1 + y2
⇒ x2 - y2 = 0
⇒ (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
⇒ 1 + x2 = y
⇒ x2 + 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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