Final answer:
The inverse function of g(x) = 2∛x - 3 + 4 is found by reversing the operations applied to x, leading to the correct answer option c. f(x) = (x - 4)^3 / 8 + 3.
Step-by-step explanation:
To find the inverse of the function g(x) = 2∛x - 3 + 4, we need to perform a series of algebraic steps that 'undo' the original function. First, replace g(x) with y to get y = 2∛x - 3 + 4.
Then, solve for x in terms of y by reversing the operations that were applied to x in the original function.
- Subtract 4 from both sides to get y - 4 = 2∛x - 3.
- Add 3 to both sides to get y - 1 = 2∛x.
- Raise both sides to the third power to eliminate the cube root, leading to (y - 1)^3 = (2∛x)^3 or (y - 1)^3 = 8x.
- Divide both sides by 8 to isolate x, resulting in x = (y - 1)^3 / 8.
- Finally, replace y with x, as we want the function in terms of x. The inverse function is f(x) = (x - 1)^3 / 8.
Comparing the given options, option c. f(x) = (x - 4)^3 / 8 + 3 is the correct choice for the inverse function as it follows the steps we've taken, but with an additional translation by +3 to account for the original -3 in g(x).