Inverse Relationships

INVERSE OF ONE-TO-ONE FUNCTION

by MAY ROSE FROGOSO

​INVERSE OF RATIONAL FUNCTION

​A relation reversing the process performed by any function f(x) is called inverse of f(x). this means that the domain of the inverse is the range of the original function and that the range of the inverse is the the original function.

​

​INVERSE OF ONE-TO-ONE FUNCTION

​let f be a one-to-one with domain A and range B. then the inverse of

​

is a function with domain B and range A defined by

if and only if f(x)=y for any y in B.

​

A function has an inverse if and only if it is one-to-one. inverting the x- and y- values of a function results in a function if and only if the original function is one-to-one.

​EXAMPLE

​Find the inverse of the function described by the set of ordered pairs {(1,-3), (2,1), (3,3), (4,5), (5,7)}

​SOLUTION.

​

​Switch the coordinates of each ordered pair.

​Original Function: {(1,-3), (2,1), (3,3), (4,5), (5,7)}

​Inverse Function: {(-3,1), (1,2), (3,3), (5,4), (7,5)}

​to find the inverse of a one-to-one function:

​a. write the function in the form y=f(x);

​b. interchange the x and y variables;

​c. solve for y in terms ofx

​EXAMPLE 1:

​Find the inverse of f(x) = 3x+1.

​SOLUTION. f(x)=3x+1

​ y= 3x+1

​ x= 3y+1

​ x-1= 3y

​

​

​

​ or

​EXAMPLE 2:

​Find the inverse of

​SOLUTION.

​EXAMPLE 3: