Question

Given f(x) = 2x³, explicitly find the inverse function and use it to determine the value of f⁻¹(16). (Recall that the cube root of a can be written as a⅓).

Answer 1

The inverse function of f(x) = 2x³ is f⁻¹(x) = (x/2)⅐. To find f⁻¹(16), we plug in 16 and solve, resulting in f⁻¹(16) being equal to 2.

Step-by-step explanation:

To find the inverse function of f(x) = 2x³, we need to swap the roles of x and y and solve for y:

1. Start with y = 2x³.
2. Rewrite it as x = 2y³ to swap x and y.
3. Divide both sides by 2, getting x/2 = y³.
4. Take the cube root of both sides to solve for y, giving us y = √(x/2) or y = (x/2)⅐.

So the inverse function is f⁻¹(x) = (x/2)⅐.

To find f⁻¹(16), we substitute 16 into the inverse function:

1. f⁻¹(16) = (16/2)⅐
2. f⁻¹(16) = (8)⅐
3. f⁻¹(16) = √8 or ¹²

The value of f⁻¹(16) is ¹² or 2.