Inverse of a function in Math

Is the Inverse of a function also a function?

Look at the function below and its inverse and see if you answer a very important aspect of inverse functions?

The questions: Is the inverse of a function necessarily a function?

In other words, is the relation below on the right also a function?

Answer

The answer is NO. The inverse of a function is always a relation but not necessarily a function.

The inverse of a function is also a function only when the original function is a one-to-one function.

Practice Problem One
f(x) = { (1,2) , (3,4) , (5,6) }
What is the inverse of this function?

Answer

Practice Problem Two
f(x) = { (33,14) , (23,15) , (11,12), (13, ,14 )}
What is the inverse of this function?

Answer

Practice Problem Three
f(x) = { (-1, 12) , (13, 114) , (15,61), ( 1, 12)}
What is the inverse of this function?

Answer

Practice Problem Four
What is the inverse of the function y = 3?

Answer

If the function you are given is not expressed as ordered pairs, but rather as an equation such as y = x + 1, we can't simply swap the domain and range to arrive at the inverse!

A different method is needed.
• Well, let's consider our initial function f(x) = x +1
• A few ordered pairs from this equation include { (0,1) , ( 1,2) , (2,3)}
• We therefore know that the inverse of this function must include the following ordered pairs { (1,0) , (2,1) , (3,2) }

Our original function took an element from the domain and added 1 to arrive at the corresponding element in the range.

The inverse of the function is doing the opposite is taking one away from element in the domain to arrive at the corresponding element in the range. Therefore, the inverse of the function must be f^-1(x) = x -1

Now, you are probably asking yourself if there is an easier method to solve for the inverse of a function, and the answer is yes!

How to solve for the inverse of a function

To find the inverse of a function simply solve for x then reverse x and y (the dependent and independent variables)!

Example
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Practice Problem Five
What is the inverse of the function y = x + 22?
Remember that f(x) and 'y' are often used interchangeably

Answer

Practice Problem Six
What is the inverse of the function y = 2x ?

Answer

Practice Problem Seven
What is the inverse of the function y = x² ?

Answer