Laura received her Master's degree in Pure Mathematics from Michigan State University, and her Bachelor's degree in Mathematics from Grand Valley State University. She has 20 years of experience teaching collegiate mathematics at various institutions.
Most of us have worked a job at some point in our lives, and we do so to make money. Consider a job where you get paid $200 a day. If you only work a fraction of the day, you get that fraction of $200. Thus, the total amount of money you make at that job is determined by the number of days you work.
Mathematically speaking, this scenario is an example of a function. A function is a relationship between two variables, such that one variable is determined by the other variable. In our example, if we let x = the number of days we work and y = the total amount of money we make at this job, then y is determined by x, and we say y is a function of x.
We've described this job example of a function in words. There are other ways to represent a function, as well. We're going to look at representing a function with a function table, an equation, and a graph. Let's get started!
A function table is a table of ordered pairs that follows the relationship, or rule, of a function. To create a function table for our example, let's first figure out the rule that defines our function. We have that each fraction of a day worked gives us that fraction of $200. Thus, if we work one day, we get $200, because 1 * 200 = 200. If we work two days, we get $400, because 2 * 200 = 400. If we work 1.5 days, we get $300, because 1.5 * 200 = 300. Are we seeing a pattern here?
To find the total amount of money made at this job, we multiply the number of days we have worked by 200. Thus, our rule is that we take a value of x (the number of days worked), and we multiply it by 200 to get y (the total amount of money made).
A function table can be used to display this rule. As you can see here, in the first row of the function table, we list values of x, and in the second row of the table, we list the corresponding values of y according to the function rule.
|x = # days worked||1||2||3||3.5||5||7.25||8|
|y = total money made||200||400||600||700||1000||1450||1600|
Sometimes function tables are displayed using columns instead of rows. When this is the case, the first column displays x-values, and the second column displays y-values.
|x = # days worked||y = total money made|
Another way to represent a function is using an equation. In this representation, we basically just put our rule into equation form. For our example, the rule is that we take the number of days worked, x, and multiply it by 200 to get the total amount of money made, y. Putting this in algebraic terms, we have that 200 times x is equal to y. In equation form, we have y = 200x.
Once we have our equation that represents our function, we can use it to find y for different values of x by plugging values of x into the equation. For example, if we wanted to know how much money you would make if you worked 9.5 days, we would plug x = 9.5 into our equation.
We see that if you worked 9.5 days, you would make $1,900. Not bad!
The last representation of a function we're going to look at is a graph. We saw that a function can be represented by an equation, and because equations can be graphed, we can graph a function. To represent a function graphically, we find some ordered pairs that satisfy our function rule, plot them, and then connect them in a nice smooth curve.
In our example, we have some ordered pairs that we found in our function table, so that's convenient! We have the points (1, 200), (2, 400), (3, 600), (3.5, 700), (5, 1000), (7.25, 1450), and (8, 1600). Let's plot these on a graph.
We see that these take on the shape of a straight line, so we connect the dots in this fashion.
We can use the graphical representation of a function to better analyze the function. For instance, with our example, we see that the function is rising from left to right, telling us that the more days we work, the more money we make. It also shows that we will earn money in a linear fashion.
All right, let's take a moment to review what we've learned. A function is a relationship between two variables, such that one variable is determined by the other variable. We can represent a function using words by explaining the relationship between the variables. We can represent a function using a function table by displaying ordered pairs that satisfy the function's rule in tabular form. A function can be represented using an equation by converting our function rule into an algebraic equation. Lastly, we can use a graph to represent a function by graphing the equation that represents the function.
It's very useful to be familiar with all of the different types of representations of a function. This knowledge can help us to better understand functions and better communicate functions we are working with to others. Make sure to put these different representations into your math toolbox for future use!
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