Question

Let $f:R\to R$ be the function defined by $f\left(x\right)=x^3+5.$ Then $f^{-1}\left(x\right)$ is

• A. $5-x$
• B. $(x+5{)}^{\frac{1}{3}}$
• C. $(x-5{)}^{\frac{1}{3}}$
• D. $(5-x{)}^{\frac{1}{3}}$

Answer 1

The correct option is C $(x-5{)}^{\frac{1}{3}}$

Find inverse of the given function.

Given: $f\left(x\right)=x^3+5,f:R\to R$

Since, $f\left(x\right)$ is bijective $\forall x\in R$

$\therefore f\left(x\right)$ is invertible.

Let $y=x^3+5$

$\Rightarrow x^3=y-5$

$\Rightarrow x=(y-5{)}^{\frac{1}{3}}$

Now, Replace $x\text{by}{f}^{-1}\left(x\right)$ and replace $y\text{by}x$

$\Rightarrow f^{-1}\left(x\right)=(x-5{)}^{\frac{1}{3}}$

Hence, the correct answer is (b)