# Properties of a function and its inverse relationship

### Functions and Their Inverses - Worked Examples

In this section we will define an inverse function and the notation this means that there is a nice relationship between these two functions. Determine the domain and range of an inverse. the original function becomes the output of the inverse function and the output of the original function becomes the input of the inverse .. Example: Testing Inverse Relationships Algebraically. Since a function is a special type of binary relation, many of the properties of an inverse function correspond to properties of converse.

Note that we really are doing some function composition here. We get back out of the function evaluation the number that we originally plugged into the composition.

So, just what is going on here? In some way we can think of these two functions as undoing what the other did to a number.

**Finding the inverse of a function**

Function pairs that exhibit this behavior are called inverse functions. This can sometimes be done with functions. In most cases either is acceptable. For the two functions that we started off this section with we could write either of the following two sets of notation.

The process for finding the inverse of a function is a fairly simple one although there are a couple of steps that can on occasion be somewhat messy. This is done to make the rest of the process easier. This is the step where mistakes are most often made so be careful with this step. This work can sometimes be messy making it easy to make mistakes so again be careful.

Most of the steps are not all that bad but as mentioned in the process there are a couple of steps that we really need to be careful with since it is easy to make mistakes in those steps. For all the functions that we are going to be looking at in this course if one is true then the other will also be true.

### Calculus I - Inverse Functions

However, there are functions they are beyond the scope of this course however for which it is possible for only one of these to be true. This is brought up because in all the problems here we will be just checking one of them. Sciencing Video Vault Inverse Mathematical Operations Addition is the most basic of arithmetic operations, and it comes with an evil twin — subtraction — that can undo what it does.

Let's say you start with 5 and you add 7. You get 12, but if you subtract 7, you'll be left with the 5 with which you started. The inverse of addition is subtraction, and the net result of adding and subtracting the same number is equivalent of adding 0.

### Examples of Inverse Relationships in Math | Sciencing

A similar inverse relationship exists between multiplication and division, but there's an important difference. The net result of multiplying and dividing a number by the same factor is to multiply the number by 1, which leaves it unchanged.

This inverse relationship is useful when simplifying complex algebraic expressions and solving equations. Another pair of inverse mathematical operations is raising a number to an exponent "n" and taking the nth root of the number.

## Inverse function

The square relationship is the easiest to consider. If you square 2, you get 4, and if you take the square root of 4, you get 2. This inverse relationship is also useful to remember when solving complex equations.

- Properties of Inverse Functions
- Functions and Their Inverses

Functions Can Be Inverse or Direct A function is a rule that produces one, and only one, result for each number you input. The set of numbers you input is called the domain of the function, and the set of results the function produces is the range. If the function is direct, a domain sequence of positive numbers that get larger produces a range sequence of numbers that also get larger.

An inverse function behaves in a different way.