Respuesta :
Answer:
Inverse of given function [tex]f\left(x\right)=x^{3}-2[/tex] is [tex]f^{-1}\left(x\right)=\sqrt[3]{x+2}[/tex]
Step-by-step explanation:
Given that [tex]f\left(x\right)=x^{3}-2[/tex] and it is one to one function.
Following are the steps to find the inverse of the above function.
Step 1: Replace [tex]f\left(x\right)[/tex] with y
[tex]y=x^{3}-2[/tex]
Step 2: Interchange x and y.
[tex]x=y^{3}-2[/tex]
Step 3: Solve for y.
Rewriting the equation in step 2,
[tex]y^{3}-2=x[/tex]
Add 2 on both sides,
[tex]y^{3}-2+2=x+2[/tex]
[tex]y^{3}=x+2[/tex]
Taking cube root on both sides,
[tex]\sqrt[3]{y^{3}}=\sqrt[3]{x+2}[/tex]
Applying radical rule,
[tex]\sqrt[n]{x^{m}}=x^{\frac{m}{n}}[/tex]
So,
[tex]\left(y^{3}\right)^{\frac{1}{3}}=\sqrt[3]{x+2}[/tex]
Simplifying,
[tex]y=\sqrt[3]{x+2}[/tex]
The resulting equation is inverse function of the given function.
[tex]\therefore f^{-1}\left(x\right)=\sqrt[3]{x+2}[/tex]
