# Examples of Inverse Relationships in Math ••• Comstock/Comstock/Getty Images

You can look at inverse relationships in mathematics in three ways. The first way is to consider operations that cancel each other out. Addition and subtraction are the two most obvious operations that behave this way.

A second way to look at inverse relations is to consider the type of curves they produce when you graph relationships between two variables. If the relationship between the variables is direct, then the dependent variable increases when you increase the independent variable, and the graph curves toward increasing values of both variables. However, if the relationship is an inverse one, the dependent variable gets smaller when the independent one increases, and the graph curves toward smaller values of the dependent variable.

Certain pairs of functions provide a third example of inverse relationships. When you graph functions that are the inverse of one another on an x-y axis, the curves appear as mirror images of each other with respect to the line x = y.

## 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.

A similar inverse relationship exists between multiplication and division. 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 ​n​th root of the number. 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.

## 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.

f(x) = 2x + 2, f(x) = x^2 \text{ and } f(x) = \sqrt{x}

are all direct functions.

An inverse function behaves in a different way. When the numbers in the domain get larger, the numbers in the range get smaller.

f(x) = \frac{1}{x}

is the simplest form of an inverse function. As x gets larger, f(​x​) gets closer and closer to 0. Basically, any function with the input variable in the denominator of a fraction, and only in the denominator, is an inverse function. Other examples include

f(x) = \frac{n}{x}

where ​n​ is any number,

f(x) = \frac{n}{\sqrt{x}}

and

f(x) = \frac{n}{x +w}

where ​w​ is a any integer.

## Two Functions Can Have an Inverse Relationship to Each Other

A third example of an inverse relationship in mathematics is a pair of functions that are inverse to each other. As an example, suppose you input the numbers 2, 3, 4 and 5 into the function

y = 2x + 1

You get these points: (2,5), (3,7), (4,9) and (5,11). This is a straight line with slope 2 and ​y​-intercept 1.

Now reverse the numbers in the brackets to create a new function: (5,2), (7,3), (9,4) and (11,5). The range of the original function becomes the domain of the new one and the domain of the original function becomes the range of the new one. It's also a line, but its slope is 1/2 and its ​y​-intercept is −1/2. Using the

y = mx + b

form of a line, you find the equation of the line to be

y = \frac{1}{2}(x - 1)

This is the inverse of the original function. You could just as easily derive it by switching ​x​ and ​y​ in the original function and simplifying to get ​y​ by itself on the left of the equal sign.

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