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.
4. Give your answer in terms of f -1 (x).

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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