Question

Find the inverse function of f(x) = 8/(3√x).

a) f⁻¹(x) = (64/27)x³
b) f⁻¹(x) = (1/64)x³
c) f⁻¹(x) = (1/27)x³
d) f⁻¹(x) = (27/8)x³

Answer 1

The inverse function of f(x) = 8/(3√x) is found by swapping x and y, manipulating algebraically, and simplifying to obtain f⁻¹(x) = (1/27)x^3, which is option c.

Step-by-step explanation:

To find the inverse function of f(x) = 8/(3√x), we need to solve for x in terms of y, where y = f(x). Let's follow these steps:

1. Swap x and y to get x = 8/(3√y).
2. Multiply both sides by 3√y to isolate the fraction on the right: 3√y × x = 8.
3. Divide by x to solve for the cube root of y: 3√y = 8/x.
4. Cube both sides to remove the cube root: (3√y)^3 = (8/x)^3.
5. Simplify the cube: 27y^3 = 512/x^3.
6. Take the reciprocal of both sides to solve for y: y = 512/(27x^3).
7. Simplify the fraction by dividing the numerator and denominator by 512: y = 1/(27x^3).

Hence, the inverse function is f⁻¹(x) = (1/27)x^3, which corresponds to option c.