Inverse Functions

Functions

Basic Conceptions

Cartesian Coordinate System

Domain and Range

Inverse Functions

Even-Odd Symmetry of Functions

Periodicity of Functions

Exponential and Logarithmic Functions

Exponential Functions

Logarithmic Functions
Natural Logarithms

Sets and Intervals

Hyperbolic Functions

Inverse Functions

Key Topics Remaining: Even-Odd Symmetry of Functions » Exponential Functions » Logarithmic Functions » Natural Logarithms

Let a function y = f ( x ) is defined by a set of ordered pairs ( x, y ).
Then the inverse relation ( y, x ) determines a function x = g ( y ) such that

y = f ( x ) = f ( g ( y ) )

and

x = g ( y ) = g ( f ( x ) ).

One can see that both composite functions, f ( g ( y ) ) and g ( f ( x ) ), represent the identical transformations of arguments. Given an argument, two rules ( f and g ) produce the argument, that is, being applied together they do nothing, they only restore the argument:

In other words, the rule g represents the inverse operation of f and vice versa.
If to interchange the variables x and y in the equation x = g ( y ) and to rename the rule g into f – 1 then we obtain the following definition of the mutually inverse functions.

Two functions, y = f ( x ) and y = f – 1( x ), are said to be inverse of each other if

f ( f – 1( x ) ) = x

and

f – 1 ( f ( x ) ) = x.

The notation of the inverse operation in form of f – 1 has its origin in the fact that

a a – 1 = b b – 1 = f f – 1 = … = 1

for any non-zero numbers a, b, f, etc.

To find the inverse function of y = f ( x ),
interchange x with y.
Then solve the obtained equation for y.

Graphics of the mutually inverse functions are symmetric with respect to the bisector of the first and third quadrants.