Also, f [ g (12)] = 12. For these functions, it can be
for any value of x . These functions are inverse functions
of each other.
Only functions that are one-to-one have inverses.
One-to-One Functions A function f is a one-to-one function if, for elements a and b from the domain of f , a b implies f ( a ) f ( b ).
Example Decide whether each function is one-to-one.
(a) For this function, two different x -values produce two different y -values.
(b) If we choose a = 3 and b = –3, then 3 –3, but
Horizontal Line Test
Example Use the horizontal line test to determine
whether the graphs are graphs of one-to-one functions.
If every horizontal line intersects the graph of a function at no more than one point, then the function is one-to-one. Not one-to-one One-to-one
are inverse functions of each other.
Let f be a one-to-one function. Then, g is the inverse function of f and f is the inverse of g if
Finding an Equation for the Inverse Function
Finding the Equation of the Inverse of y = f ( x ) For a one-to-one function f defined by an equation y = f ( x ) , find the defining equation of the inverse as follows. ( Any restrictions on x and y should be considered.) 1. Interchange x and y. 2. Solve for y. 3 . Replace y with f -1 ( x ).
Example of Finding f -1 ( x )
Example Find the inverse, if it exists, of
Write f ( x ) = y . Interchange x and y . Solve for y . Replace y with f -1 ( x ).
The Graph of f -1 ( x )
f and f -1 ( x ) are inverse functions, and f ( a ) = b for real numbers a and b . Then f -1 ( b ) = a .
If the point ( a , b ) is on the graph of f , then the point ( b , a ) is on the graph of f -1 .
If a function is one-to-one, the graph of its inverse f -1 ( x ) is a reflection of the graph of f across the line y = x .
Finding the Inverse of a Function with a Restricted Domain
Solution Notice that the domain of f is restricted
to [ – 5, ), and its range is [0, ). It is one-to-one and
thus has an inverse.
The range of f is the domain of f -1 , so its inverse is
Important Facts About Inverses
If f is one-to-one, then f -1 exists.
The domain of f is the range of f -1 , and the range of f is the domain of f -1 .
If the point ( a , b ) is on the graph of f , then the point ( b , a ) is on the graph of f -1 , so the graphs of f and f -1 are reflections of each other across the line y = x .
Application of Inverse Functions
Example Use the one-to-one function f ( x ) = 3 x + 1 and the
numerical values in the table to code the message BE VERY CAREFUL.
A 1 F 6 K 11 P 16 U 21
B 2 G 7 L 12 Q 17 V 22
C 3 H 8 M 13 R 18 W 23
D 4 I 9 N 14 S 19 X 24
E 5 J 10 O 15 T 20 Y 25
Solution BE VERY CAREFUL would be encoded as
7 16 67 16 55 76 10 4 55 16 19 64 37
because B corresponds to 2, and f (2) = 3(2) + 1 = 7,