F⁻¹(x)

Definition

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 you to find the input value given the output value.

5 Must Know Facts For Your Next Test

1. The inverse function f⁻¹(x) undoes the operation performed by 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 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).
4. Radical functions, such as square root and cube root functions, can be inverted using the inverse function f⁻¹(x).
5. Graphically, the graph of the inverse function f⁻¹(x) is the reflection of the graph 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) undoes the operation performed by 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. For a function to have an inverse, it must be one-to-one, where each output value is associated with only one input value.
• Describe how the domain and range of a function and its inverse function are related.
  • 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). This is because the inverse function reverses the input and output values of the original function. If the original function has a restricted domain or range, the inverse function will have a corresponding restricted range or domain, respectively.
• Analyze the relationship between the graphs of a function f(x) and its inverse function f⁻¹(x).
  • The graph of the inverse function f⁻¹(x) is the reflection of the graph of the original function f(x) across the line y = x. This is because the input and output values of the original function and its inverse function are reversed. The x-coordinates of the original function become the y-coordinates of the inverse function, and vice versa. This visual representation helps illustrate the relationship between a function and its inverse.

Related terms

Inverse Function:

A function that reverses the operation of the original function, such that if f(x) = y, then f⁻¹(y) = x.

Domain and Range:

The set of input values (domain) and output values (range) for a function and its inverse function.

Radical Functions: Functions that involve roots or radicals, such as square roots, cube roots, and higher-order roots, which can be inverted using the inverse function.