 # Inverse Functions

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 f if for all x in the domain of g; for all x in the domain of f.

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 f ( f-1( y)) = y for all y in the domain of f-1

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

## Graphing Inverses :

If the graph of f is the set of points (x, y), then the graph
of f-1is the set of points ( y, x).
Since, points (x, y) and ( y, x) are symmetric with
respect to the line y = x then

 the graphs of f and f -1 are symmetric with respect to 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:

1. Write y = f (x).
2. Solve the equation for x : x = f-1 ( y)
3. Interchange x and y.

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 is one-to-one.

Find f-1(x) and its domain and range.

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