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Mathematics · 2023 · 5 marks
CBSE 2023 · Region 3 · Set 1 · Q32
Show that a function f: R → R defined as f ( x )=(5 x -3)/4 is both one-one and onto.
Show that a function $\displaystyle \mathrm{f}: \mathbb{\mathrm{R}} \rightarrow \mathbb{\mathrm{R}}$ defined as $\displaystyle \mathrm{f}(\mathrm{x})=\frac{5 \mathrm{x}-3}{4}$ is both one-one and onto.
Marking-scheme solution
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Let $\displaystyle \mathrm{x}_{1}, \mathrm{x}_{2} \in \mathrm{R}$ (Domain) such that $\displaystyle \mathrm{f}\left(\mathrm{x}_{1}\right)=\mathrm{f}\left(\mathrm{x}_{2}\right)$ $\displaystyle \Rightarrow \frac{5 \mathrm{x}_{1}-3}{4}=\frac{5 \mathrm{x}_{2}-3}{4}$
$$\Rightarrow 5 \mathrm{x}_{1}=5 \mathrm{x}_{2} \Rightarrow \mathrm{x}_{1}=\mathrm{x}_{2}
$$ $\displaystyle \therefore \mathrm{f}$ is one-one.
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