Final answer:
To determine the value of (f-1)'(8), we need to find the derivative of the inverse of the given function f(x). By finding the inverse function and its derivative, we can evaluate it at x = 8. The value of (f-1)'(8) is ±√41/41.
Step-by-step explanation:
In order to determine the value of (f-1)'(8), we need to find the derivative of the inverse of the function f(x) = x² - 3x at the point x = 8. Let's start by finding the inverse of f(x).
To find the inverse of f(x), we can switch the roles of x and y in the equation y = x² - 3x and solve for x in terms of y.
y = x² - 3x --> x² - 3x - y = 0
Using the quadratic formula, we can find that the inverse function is given by f-1(x) = (3 ± √(9 + 4y))/2.
Next, we can find the derivative of f-1(x) and evaluate it at x = 8.
f-1(x) = (3 ± √(9 + 4y))/2 --> (f-1)'(x) = ± 2/(2√(9 + 4y))
Since we are evaluating at x = 8, we substitute 8 into the equation:
(f-1)'(8) = ± 2/(2√(9 + 4(8))) = ± 2/(2√41)
We can simplify further by rationalizing the denominator:
(f-1)'(8) = ± 2/(2√41) * (√41/√41) = ± √41/41