F^(-1)(x)

Definition

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), results in the original input value. The inverse function allows you to 'undo' the original function, reversing the transformation and recovering the original input.

5 Must Know Facts For Your Next Test

1. The inverse function f^(-1)(x) 'undoes' the original function f(x), so that f^(-1)(f(x)) = x.
2. For a function to have an inverse, it must be one-to-one, meaning each output value is associated with only one input value.
3. The graph of the inverse function f^(-1)(x) is the reflection of the graph of f(x) across the line y=x.
4. Finding the inverse function involves interchanging the x and y variables in the original function equation.
5. Inverse functions are useful for solving equations involving the original function, as f^(-1)(f(x)) = 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). If f(x) = y, then f^(-1)(y) = x. This means that the input and output values of the original function and its inverse function are reversed. Graphically, the inverse function is the reflection of the original function across the line y=x.
• Describe the properties a function must have in order to have a unique inverse function.
  • For a function f(x) to have a unique inverse function f^(-1)(x), the function must be one-to-one. This means that each output value of f(x) 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 f(x) more than once. This ensures that the inverse function f^(-1)(x) is also a function, with each input value corresponding to a unique output value.
• Explain how to find the inverse function f^(-1)(x) given the original function f(x).
  • To find the inverse function f^(-1)(x) of a given function f(x), you can follow these steps: 1) Interchange the x and y variables in the original function equation, so that y = f(x) becomes x = f(y). 2) Solve this equation for y in terms of x, which will give you the equation for the inverse function f^(-1)(x). 3) Verify that the inverse function satisfies the property f^(-1)(f(x)) = x, ensuring that it correctly 'undoes' the original function.

Related terms

Inverse Function: A function that 'undoes' the original function, so that if f(x) = y, then f^(-1)(y) = x.

One-to-One Function: A function where each output value is associated with only one input value, allowing for a unique inverse function.

Reflection Across the Line y=x: The geometric interpretation of finding the inverse function, where the graph of f(x) is reflected across the line y=x to obtain the graph of f^(-1)(x).