The given function is
f(x) = x^3 - 1
We want to find the inverse of the function, f^-1(x). The first step is th replace f(x) with y. It becomes
y = x^3 - 1
The next step is to interchange x and y and solve for y. We have
x = y^3 - 1
Adding 1 to both sides of the equation, we have
x + 1 = y^3 - 1 + 1
y^3 = x + 1
Taking the cube root of both sides of the equation, we have
[tex]\begin{gathered} y\text{ = }\sqrt[3]{x\text{ + }1} \\ \text{Put y = f}^{-1}(x),\text{ we have} \\ f^{-1}(x)\text{ = }\sqrt[3]{x\text{ + }1} \end{gathered}[/tex]
To plot the graph of the the inverse of the function, we would substitute values of x into the function and find corresponding values of y. These x and y values would be plotted on the graph. We have
[tex]\begin{gathered} \text{if x = 0, y = }\sqrt[3]{0\text{ + 1}}\text{ = }\sqrt[3]{1}\text{ = 1} \\ \text{If x = 7, y = }\sqrt[3]{7\text{ + 1}}\text{ = }\sqrt[3]{8}\text{ = 2} \\ \text{if }x\text{ = 26, y = }\sqrt[3]{26\text{ + 1}}\text{ = }\sqrt[3]{27}\text{ = 3} \end{gathered}[/tex]
The graph is shown below
