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RELATIONS AND FUNCTIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsRELATIONS AND FUNCTIONS1 Mark
MCQ
Choose the correct answer from the given four options. Let $f : R \rightarrow R$ be the functions defined by $f(x) = x^3 + 5.$ Then $f^{-1}(x)$ is :
  • A
    $(\text{x}+5)^\frac{1}{3}$
  • ✓
    $(\text{x}-5)^\frac{1}{3}$
  • C
    $(5-\text{x})^\frac{1}{3}$
  • D
    $5-\text{x}$
✓

Answer

Correct option: B.
$(\text{x}-5)^\frac{1}{3}$
we are given that, $\text{f}(\text{x})=\text{x}^3 +5$
Let us suppose, $\text{y}=\text{x}^3+5$
$\Rightarrow\ \text{x}^3=\text{y}-5$
$\Rightarrow\text{x}=(\text{y}-5)^{\frac{1}{3}}$
$\begin{bmatrix}\because\text{f}(\text{x})=\text{y}\\\Rightarrow\text{x}=\text{f}^{-1}(\text{y})\end{bmatrix}$
$\Rightarrow\text{f}^{-3}(\text{y})=(\text{y}-5)^{\frac{1}{3}}$
$\Rightarrow\text{f}^{-1}(\text{x})=(\text{x}-5)^{\frac{1}{3}}$

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