# How Functions Can Increase Your Income by Still Working the Same Number of Hours!

Updated: Jul 31, 2018

What if I told you that understanding how functions work can earn you more money for the same amount of hours you put in? Would you believe it if I told you? Well, it is possible, but first you will need to know how functions work and how we can incorporate them into our daily life, such as our finances.

In mathematics, functions are the relationships between a set of inputs (domain) and a set of outputs (range). Depending on the input you give a function, it will generate an output for that specific input. Therefore, functions can be thought of as output = (input given), or mathematically its simply written as f(x) = x. In more detail, that equation means that for any number inputted into variable x, you will get the same number for the output f(x). But what happens if we change the expression to f(x) = x + 2? This will change everything because now if we input any number into x, the output will be a different number than the one initially inputted. For example, f(x) = (2) + 2 will give us 4 as the output, and not 2. Moreover, you also need to understand that only one specific range can be mapped to one domain, otherwise it is not considered a function. For example, if we set f(1) = 1 and f(1) = 2, then domain x is now linked to 1 and 2 which is not valid, and there are methods to determine if a function is valid, such as the vertical line test.

Now that you have learned the concept of a function and its mathematical notation, let us apply this knowledge in terms of your finances. So, we have actually already mentioned a function already in the introduction -- that is, that the amount of income you earn is based off a certain number of hours worked. Let us look at this from a mathematical viewpoint and form a standard notation of a function. First, let the number of hours worked be H and it will be considered the input or domain of the function because the amount of income earned is completely based off of this number and so it is also considered the independent variable; next, let amount earned be the output or range of the function because the number outputted will completely depend on the number of hours you have worked and it is the dependent variable and lastly we shall label it C. Since the amount earned is based off the salary per hour and the number of hours worked, let us multiply H by some salary earned per hour ($50). Now we can finally formulate our final relationship: f(H) or C = H * 50. That can now allow us to calculate in a mathematical sense someone's income per week based on the number of hours worked and that they are paid $50 per hour. For example, let us input forty hours, and see what we get. f(40) = 40 * 50, = $2000. The person would earn $2000 a week for working forty hours and he is paid $50 the hour. Now that we have explored the relationship of income and hours, let us see how we can increase C by changing the equation, but still keeping the same number of hours inputted.

We will now be adding more variables to the same function's expression earlier to see how we can better increase the outputted amount, C. Adding more variables to a function is called adding terms. Terms are groups of a coefficient and a variable raised to a power. Functions that contain only one term are called monomial, two terms is binomial, and more than three are called polynomial functions. For example, a monomial is f(x) = x^2, a binomial is f(x) = x^2 + y^4, and a polynomial function is f(x) = x^2 + y^4 + z^3. Terms are separated by either a positive or negative sign. A special type of polynomial function is a quadratic function and it is a function of the second-degree. Quadratic formulas usually take the form of Ax^2 + Bx + C = 0. A degree of a function is the term's highest exponent. For example, let us look at a regular quadratic: f(x) = x^2 + 5x+ 5. You will notice that the highest exponent is 2, then 1, then 0 (no variable means the power is 0). Although quadratic equations are very interesting and more complicated, they will not be necessary for this financial management, and it will be too long of an article to further introduce it. We will stick to the simple elements of a function, and work from there.

Now that I have introduced you to the concept of a function, basic terms, and quadratic equations, we can now finally incorporate that into our final equation. Let us take the original equation (f(H) or C = H * 50) with H = 40 and let us if we can modify or add any more variables to the equation to improve the amount earned. So the solution would be to efficiently somehow decrease the amount of time needed to complete the work per hour, by let us say half, and so that would likewise increase the salary but working the same number of hours! Let us represent this mathematically: C = (H * 50) / 2. To break it down, since we are essentially working half the time per week (but still being completing the work) we will divide the original equation by half. We will keep the multiplication of the hourly salary the same, since that didn't change. Now if we input 40 hours into the new equation, we shall see a half decrease in C or income earned: f(40) = (40 * 50) / 2, = 1000. But how did we earn more money? Well, since the work was completed in half the time, the person could work a second job that pays more for the remainder of the 20 hours per week, and earn more income by the end of the week in total. For example, let us say that the person found another job that pays twice her first job and he has to complete the job in 20 hours. Let us input this into the function: f(H) = (H / 2) * 50 + (20 * 100). And let us input the 40 hours into the function now f(40) = (40 / 2) * 50 + (20 * 100), = $3000. Now the person has earned a $1000 more but still working the same number of hours (40)!

Finally, now that you have learned the concept of a function, its mathematical notation, basic terms, and what a quadratic function is, you may now apply this knowledge in different aspects of life, for example, this article specifically targets finances but you can apply it in other places as well. Functions have helped us to mathematically represent the amount a person earns based of off the number of hours they worked per week. Moreover, they have helped us to potentially increase their income (C) by still working the same number of hours per week.