Continuity
Inverse functions
Objectives
- Verify inverse functions
- Determine the domain and range of an inverse function and restrict the domain of a function to make it one-to-one.
- Find or evaluate the inverse of a function.
- Use the graph of a one-to-one function to graph its inverse function on the same axes.
Summary
One to One:
If no two elements in the domain of f correspond to the same element in the range of f then the functions is one to one.
Figure 5 Each element of the domain corresponds exactly to one element in the range of a function \(f\) thus being one to one.
Inverse Function:
For any one-to-one function \(f(x) = y\), a function \(f^{-1} (x)\) is an inverse function of \(f\) if \(f^{-1}(y) = x\).
How to test if two function \(f(x)\) and \(g(x)\) are inverses of each other:
- Determine where \(f(g(x)) = x\) or \(g(f(x)) = x\)
- If either statement is true, then both are true, and \(g = f^{-1}\) and \(f = g^{-1}\). If either statement is false, then both are false.
Domain and Range of Inverse Function:
The range of the original function \(f(x)\) is the domain of the inverse function \(f^{-1}(x)\).
The domain of the original function \(f(x)\) is the range of the inverse function \(f^{-1}(x)\)
If you are given a formula, to find the inverse you must:
- Make sure the function is a one-to-one function
- Solve for \(x\).
- Interchange \(x\) and \(y\).