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RELATIONS AND FUNCTIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsRELATIONS AND FUNCTIONS3 Marks
Question
Classify the following functions as injection, surjection or bijection:
f : R → R, defined by f(x) = sinx
✓

Answer

f : R → R, given by f(x) = sinx

Injective: Let $\text{x, y}\in\text{R}$ such that

f(x) = f(y)

⇒ sinx = siny

$\Rightarrow\ \text{x}=\text{n}\pi+(-1)^{\text{n}}\text{y}$

$\Rightarrow\ \text{x}\neq\text{y}$

$\therefore$ f is not one-one.

Surjective: Let $\text{y}\in\text{R}$ be arbitrary such that

f(x) = y

⇒ sinx = y

⇒ x = sin-1y

Now, for $\text{y}>1\times\notin\text{R}$ (domain)

$\therefore$ f is not onto.

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