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Functions — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsFunctions5 Marks
Question
Classify the following functions as injection, surjection or bijection:
$f : R \rightarrow R,$ defined by $f(x) = x^3 - x$
✓

Answer

$f : R \rightarrow R,$ defined by $f(x) = x^3 - x$
Injective: Let $\text{x, y}\in\text{R}$
such that, $f(x) = f(y)$
$\Rightarrow x^3 - x = y^3 - y $
$\Rightarrow x^3 - y^3 - (x - y) = 0 $
$\Rightarrow (x - y)(x^2 + xy + y^2 - 1) = 0$
$\because\ \text{x}^2+\text{xy}+\text{y}^2\geq0$
$\Rightarrow\ \text{x}^2+\text{xy}+\text{y}^2-1\geq-1$
$\therefore\ \text{x}^2+\text{xy}+\text{y}^2-1\neq0$
$\Rightarrow \text{x}-\text{y}=0\Rightarrow \text{x}=\text{y}$
$\therefore$ f is one-one.
Surjective: Let $\text{y}\in\text{R},$
then$ f(x) = y $
$\Rightarrow x^3 - x - y = 0$
We know that a degree 3 equation has atleast one real solution.
Let $\text{x}=\alpha$ be that real solution
$\therefore\ \alpha^2-\alpha=\text{y}$
$\Rightarrow\ \text{f}(\alpha)=\text{y}$
$\therefore$ For each $\text{y}\in\text{R,}$
there exist $\text{x}=\alpha\in\text{R}$
such that $\text{f}(\alpha)=\text{y}$
$\therefore f$ is onto.

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