Relations and Functions — MATHS STD 12 Science — Question
Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSRelations and Functions3 Marks
Question
A function $f: R \rightarrow R$ given by $f(x)=2 x+3$. Prove that $f$ is invertible.
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Answer
$f: R \rightarrow R$ and $f(x)=2 x+3$
suppose $x_1, x_2 \in R$ such that $f\left(x_1\right)=f\left(x_2\right)$
$\Rightarrow 2 x_1+3=2 x_2+3$
$\Rightarrow 2 x_1=2 x_2$
$\Rightarrow x_1=x_2$
$\therefore f$ is one$-$one.
Again suppose $y=2 x+3$
$\therefore x=\frac{y-3}{2}$
now $f\left(\frac{y-3}{2}\right)=2\left(\frac{y-3}{2}\right)+3=y-3+3=y$
$\Rightarrow f(x)=y$
hence function is onto.
$f$ is one$-$one onto hence $f$ is invertible so, $f^{-1}$ will exist.
$f^{-1}(y)=x=\frac{y-3}{2}$
$f^{-1}(x)=\frac{x-3}{2}$ Hence Proved.
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