Let function R → R is defined as f(x) = 2x³ – 1, then ‘f’ is

The correct option is [(1+x)/2]¹/³ .

To determine the inverse of the given function, we need to express the variable $x$ in terms of $y$.

Let the given function be represented as:

$$y=f\left(x\right)=2x^3-1$$

Now, we solve for $x$ step-by-step:

First, add 1 to both sides of the equation:

$$y+1=2x^3$$

Next, divide both sides by 2 to isolate the cubic term:

$$x^3=\frac{y+1}{2}$$

To solve for $x$, take the cube root of both sides of the equation:

$$x={\left(\frac{y+1}{2}\right)}^{1/3}$$

Since $y=f\left(x\right)$, we have $x=f^{-1}\left(y\right)$. Substituting this gives:

$$f^{-1}\left(y\right)={\left(\frac{1+y}{2}\right)}^{1/3}$$

Replacing the dummy variable $y$ with $x$ to write it in standard function notation, we obtain the inverse function:

$$f^{-1}\left(x\right)={\left[\frac{1+x}{2}\right]}^{1/3}$$

Therefore, the inverse function of $f$ is indeed ${\left[\frac{1+x}{2}\right]}^{1/3}$.