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RELATIONS AND FUNCTIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsRELATIONS AND FUNCTIONS3 Marks
Question
Let $f : R \rightarrow R$ be the function defined by $f(x) = 2x – 3 ∀ x \in R$. write $ f^{-1}.$
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Answer

Given $f(x) = 2x - 3 ∀ x \in R$ Now,
Let $a, b \in R$ such that
$f(a) = f(b) \Rightarrow 2a - 3 = 2b - 3\Rightarrow a = b \Rightarrow f(x)$ is One-One.
Also, If $x, y \in R$ such that $f(x) = y \Rightarrow 2x - 3 = y$
$\Rightarrow\ \text{x}=\frac{\text{y}+3}{2}=\text{g}(\text{y})\ \forall\ \text{y}\in\text{R}$
$\Rightarrow f(x)$ is Onto and therefore is bijective implies $f(x)$ has an inverse
Let $f^{-1}$ denote the inverse of $f(x)$
then,$\text{f}^{-1}\text{x}=\text{g}(\text{x})=\frac{\text{x}+3}{2}\ \forall\ \text{x}\in\text{R}$

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