Verify that f has an inverse and find (f^-1)^'(c)....

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Key Concepts

One-to-One Functions

A function must be one-to-one (injective) to have an inverse, meaning that each output is produced by exactly one input. This is typically confirmed by showing that the function is strictly increasing or strictly decreasing over its domain, ensuring no two different inputs yield the same output.

Inverse Function Theorem

The Inverse Function Theorem provides conditions under which a function has a differentiable inverse. Specifically, if a function is continuously differentiable and its derivative is nonzero at a certain point, then the inverse function exists near that point and is differentiable. The derivative of the inverse function at a point can be computed as the reciprocal of the derivative of the original function at the corresponding point.

Derivative of the Inverse Function

For a differentiable function that has an inverse, the derivative of the inverse function at a particular value is given by the formula (f?¹)'(y) = 1 / f'(x), where x and y are related by y = f(x). This formula is essential when we need to compute the rate of change of the inverse function without explicitly finding its analytical form.

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