The inverse function of a function f(x) is denoted as f^(-1)(x). It represents the function that undoes the operation performed by the original function f(x), allowing the input to be recovered from the output.
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The inverse function f^(-1)(x) is read as 'f inverse of x' and is the function that undoes the operation of the original function f(x).
For a function f(x) to have an inverse function, it must be a one-to-one function, meaning each output value is associated with only one input value.
The domain of the inverse function f^(-1)(x) is the range of the original function f(x), and the range of f^(-1)(x) is the domain of f(x).
To find the inverse function f^(-1)(x), you can solve the equation f(x) = y for x, then replace f(x) with y and x with f^(-1)(y).
Graphically, the inverse function f^(-1)(x) is the reflection of the original function f(x) across the line y = x.
Review Questions
Explain the relationship between a function f(x) and its inverse function f^(-1)(x).
The inverse function f^(-1)(x) is the function that undoes the operation performed by the original function f(x). This means that if you apply the original function f(x) and then apply the inverse function f^(-1)(x), you will end up with the original input value. Mathematically, this can be expressed as f(f^(-1)(x)) = x and f^(-1)(f(x)) = x, indicating that the inverse function reverses the operation of the original function.
Describe the conditions required for a function f(x) to have an inverse function f^(-1)(x).
For a function f(x) to have an inverse function f^(-1)(x), the function must be a one-to-one function. This means that each output value of the function is associated with only one input value. In other words, the function must pass the horizontal line test, where no horizontal line intersects the graph of the function more than once. If the function is not one-to-one, it will not have a unique inverse function, and additional steps may be required to find a suitable inverse function.
Explain how the domain and range of a function f(x) are related to the domain and range of its inverse function f^(-1)(x).
The domain and range of a function f(x) and its inverse function f^(-1)(x) are interchanged. The domain of the original function f(x) becomes the range of the inverse function f^(-1)(x), and the range of f(x) becomes the domain of f^(-1)(x). This is because the inverse function reverses the operation of the original function, effectively switching the input and output values. Understanding the relationship between the domain and range of a function and its inverse is crucial when working with inverse functions.