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RELATIONS AND FUNCTIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsRELATIONS AND FUNCTIONS1 Mark
MCQ
Let $\text{f}:\text{R}-\Big\{-\frac{4}{3}\Big\}\rightarrow\text{R}$ be a function defined as $f(\text{x})=\frac{4\text{x}}{3\text{x}+4}.$ The inverse of f is the map g: Range $\text{f}:\text{R}-\Big\{-\frac{4}{3}\Big\}\rightarrow\text{R}$ given by:
  • A
    $\text{g}(\text{y})=\frac{3\text{y}}{3-4\text{y}}$
  • ✓
    $\text{g}(\text{y})=\frac{4\text{y}}{4-3\text{y}}$
  • C
    $\text{g}(\text{y})=\frac{4\text{y}}{3-4\text{y}}$
  • D
    $\text{g}(\text{y})=\frac{3\text{y}}{4-3\text{y}}.$
✓

Answer

Correct option: B.
$\text{g}(\text{y})=\frac{4\text{y}}{4-3\text{y}}$
Given: $\text{f}:\text{R}-\Big\{-\frac{4}{3}\Big\}\rightarrow\text{R}\ \text{and}\ f(\text{x})=\frac{4\text{x}}{3\text{x}+4}$
Now, Range of $\text{f}\rightarrow\text{R}-\Big\{-\frac{4}{3}\Big\}$
Let $\text{y}=f(\text{x})\ \ \ \therefore\ \text{y}=\frac{4\text{x}}{3\text{x}+4}\ \ \Rightarrow\ 3\text{xy}+4\text{y}=4\text{x}$
$\Rightarrow\ \ \ \text{x}(4-3\text{y})=4\text{y}\ \ \Rightarrow\ \text{x}=\frac{4\text{y}}{4-3\text{y}}$
$\therefore\ \ f^{-1}(\text{y})=\text{g(y)}=\frac{4\text{y}}{3-4\text{y}}$
Therefore, option (B) is correct.

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