Final answer:
The statement that F^-1 (x) = -3 + cubed square root x + 4 is the inverse function of f(x) = (x - 3 )^3 + 4 is false. The correct inverse would simply be the cubed square root of (x - 4), without additional terms.
Step-by-step explanation:
Is F-1 (x) = -3 + cubed square root x + 4 the inverse function of f(x) = (x - 3)3 + 4? This statement is true. To verify, we can take the proposed inverse function and substitute it into the original function. If it returns the input of the inverse function (x), the statement is correct.
Let's substitute F-1(x) into f(x):
f(F-1(x)) = f(-3 + cubed square root x + 4) = ((-3 + cubed square root of x + 4) - 3)3 + 4 = (cubed square root of x)3 + 4 = x + 4
Since the result is not x but x + 4, we have to reconsider our initial verification. A closer look reveals that the correct inverse should simply be cubed square root of x without additions or subtractions, therefore the initial statement is actually false.
The correct inverse function should be defined as F-1(x) = cubed square root (x - 4).
Understanding inverse functions involves applying operations that 'undo' each other. For example, squaring a number is the inverse of taking the square root, just as multiplication is the inverse of division.