Suppose that the function f and its derivative with respect to x have the following values at x=0, 1, 2, 3, and 4.
Assuming the inverse function f^(-1) is differentiable, find the slope of f^(-1)(x) at
c. x=3

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Identify the value of \(c\) for which you need to find the slope of the inverse function \(f^{-1}(x)\). Here, \(c = 3\).

Recall the formula for the derivative of the inverse function: \(\left(f^{-1}\right)'(c) = \frac{1}{f'(f^{-1}(c))}\).

Find \(f^{-1}(3)\) by looking for the \(x\) value such that \(f(x) = 3\). From the table, \(f(0) = 3\), so \(f^{-1}(3) = 0\).

Use the value of \(f^{-1}(3)\) to find \(f'(f^{-1}(3)) = f'(0)\). From the table, \(f'(0) = \frac{4}{3}\).

Calculate the slope of the inverse function at \(x=3\) using the formula: \(\left(f^{-1}\right)'(3) = \frac{1}{f'(0)} = \frac{1}{\frac{4}{3}}\).

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주요 개념

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Inverse Function and Its Differentiability

An inverse function f^(-1) reverses the effect of the original function f, such that f(f^(-1)(x)) = x. For the inverse to be differentiable at a point, the original function must be continuous and have a non-zero derivative at the corresponding point. This ensures the inverse function's slope exists and can be calculated.

Derivative of the Inverse Function

The derivative of the inverse function at a point x = c is given by (f^(-1))'(c) = 1 / f'(f^(-1)(c)). This formula relates the slope of the inverse function to the slope of the original function at the corresponding point. It requires finding the input value where f(x) = c and then using the derivative of f at that input.

Using Tabulated Values to Find the Slope

Given a table of values for x, f(x), and f'(x), you can find the slope of the inverse function at x = c by first locating the x-value where f(x) = c. Then, use the corresponding f'(x) value to compute the inverse slope using the formula. This approach is practical when explicit functions are not provided.

Suppose that the function f and its derivative with respect to x have the following values at x=0, 1, 2, 3, and 4.Assuming the inverse function f^(-1) is differentiable, find the slope of f^(-1)(x) atc. x=3