Choose the correct answer from the given four options. Let f: R R… - Vidyadip

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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