Recall: The inverse of a function f is a function itself if and only if f is one-to-one.
|Let f be a one-to-one function. Then g is the
inverse function of
If g is the inverse function of f, then we write g as f-1(x) and read: “f-inverse”.
Example: Determine whether the following functions are inverses of each other:
Cancellation Rules for Inverses
|The inverse functions undo each other with
respect to the compositions:
f-1( f (x)) =
x for all x in the domain of f
Equivalent Form of the Cancellation Rules:
|f (x) = y||<->||f-1( y) = x|
|(x in domain of f)||(y in the domain of f-1)|
Note on the Domains and Ranges of the Inverses:
|Domain of f-1 = Range of f
Range of f-1 = Domain of f
|the graphs of f and f -1 are symmetric with
the line y = x.
Example: Given the graph of y = f (x). Draw the graph of its inverse.
Finding the Inverse of a One-to-one Function f:
Note: Consider all restrictions on the variables.
Example: Find f-1(x), if it is possible.
Finding the Inverse of a Domain-restricted Function:
(a) Find the inverse of
(b) The function
Find f-1(x) and its domain and range.