Inverse Functions

1. Inverse Functions We can think of an inverse function as something that “undoes” what the function itself does. For example adds 3 to the input value x. To “undo” adding 3 we would want to subtract 3. This gives us

2. One-to-One Functions Before finding the inverse of a function we need to know which functions have inverses. The functions with inverses are those which are one-to-one (1-1) A one-to-one function is a function with the characteristic that for any a and b in its domain f(a) = f(b) if and only if a = b. Another way of thinking of this is that each output value is associated with only one input value

3. More on One-to-One Functions One-to-one functions are either increasing everywhere on their domain or they are decreasing everywhere on their domain. Not a one-to-one function. Is a one-to-one function.

4. Finding Inverse Functions Rewrite the function in the formy=function rule Swap x’sandy’s Solve the new equation fory in terms of x This solution is your inverse function

5. More Finding Inverse Functions Find the inverse of the following functions (click mouse to see answer).

6. More Finding Inverse Functions Find the inverse of the following function (click mouse to see answer).

7. Graph of a Function and Its Inverse The red curve is the graph of the function and the blue curve is the graph of the inverse function Notice the symmetry about y=x