The inverse function of a function f(x) is denoted as f^(-1)(x). It represents the function that, when applied to the output of f(x), returns the original input value. The inverse function is a way to 'undo' the original function, allowing us to solve for the input variable when given the output.
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The inverse function f^(-1)(x) is only defined if the original function f(x) is one-to-one, meaning it has a unique output for each input value.
To find the inverse function f^(-1)(x), you can solve the equation f(x) = y for x in terms of y, and then replace y with x.
The derivative of the inverse function f^(-1)(x) is given by the formula $\frac{d}{dx}f^{-1}(x) = \frac{1}{\frac{d}{dx}f(f^{-1}(x))}$.
Inverse functions are often used to solve equations involving the original function, by applying the inverse function to both sides of the equation.
The domain of f^(-1)(x) is the range of f(x), and the range of f^(-1)(x) is the domain of f(x).
Review Questions
Explain the relationship between a function f(x) and its inverse function f^(-1)(x).
The inverse function f^(-1)(x) 'undoes' the original function f(x), such that if y = f(x), then x = f^(-1)(y). This means that the input and output of the original function are swapped in the inverse function. Graphically, the graph of f^(-1)(x) is the reflection of the graph of f(x) across the line y = x.
Describe the conditions under which a function f(x) has an inverse function f^(-1)(x).
For a function f(x) to have an inverse function f^(-1)(x), the original function must be one-to-one, meaning it has a unique output for each input value. If a function is not one-to-one, it will not have a well-defined inverse function. Additionally, the domain of f^(-1)(x) must be the range of f(x), and the range of f^(-1)(x) must be the domain of f(x).
Explain how the derivative of the inverse function f^(-1)(x) is related to the derivative of the original function f(x).
The derivative of the inverse function f^(-1)(x) is given by the formula $\frac{d}{dx}f^{-1}(x) = \frac{1}{\frac{d}{dx}f(f^{-1}(x))}$. This means that the derivative of the inverse function is the reciprocal of the derivative of the original function, evaluated at the point where the input and output of the original function are swapped. This relationship allows us to find the derivative of the inverse function using the derivative of the original function.