Question

What is the inverse of $f(x) = x³ - 2$?

A) $f⁻¹(x) = \sqrt[3]{x + 2}$
B) $f⁻¹(x) = x² - 2$
C) $f⁻¹(x) = \sqrt[3]{x} - 2$
D) $f⁻¹(x) = x{¹/³} - 2$

Answer 1

The inverse of the function f(x) = x³ - 2 is f⁻¹(x) = (x + 2)¹/³.

Step-by-step explanation:

The inverse of the function f(x) = x³ - 2 can be found by switching the roles of x and y and solving for y. Let's call the inverse function f⁻¹(x). The step-by-step process is as follows:

1. Replace f(x) with y.
2. Swap x and y to get x = y³ - 2.
3. Solve for y by adding 2 to both sides: x + 2 = y³.
4. To isolate y, take the cube root of both sides: (x + 2)¹/³ = y.

Therefore, the inverse of the function is f⁻¹(x) = (x + 2)¹/³. Therefore, the correct answer is A) f⁻¹(x) = (x + 2)¹/³.