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Introduction to Functions - Concept

Teacher/Instructor Carl Horowitz
Carl Horowitz

University of Michigan
Runs his own tutoring company

Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

In mathematics, a relationship describles one quantity in terms of another. A function is a type of relationship in which for each first component there is one and only one second component. In mathematics, an introduction to functions and how to identify whether or not a relationship is a function is very important building block since a lot of complex topics in upper-level math involve functions.

We're going to talk today about relationships and functions. And relationship is anything that describes one quantity in terms of another. So, say you go to the gas station. Gas is $3 a gallon, you buy 4 gallons you pay $12. That's a relationship, okay. Something goes in another thing comes out. Gas goes in, money comes out.
We're going to talk about more specifically is functions which is a relationship where for each of the first component there is one and only one of this second. So what that means is you put that three, you go to a gas station, you put in that four gallons, $3 are gone. Cost $12. If you go to a, your friend goes right at the same time and gets that same amount of gas, they would expect to pay the same amount. It doesn't make any sense that if you put in three gallons one person will pay will pay one amount, one person will pay another. So for each thing that goes in, the same thing has to come out. Same amount of gas, same amount of money has to be paid.
Okay, so with that there is actually some language, okay? So, the first component and what I mean by that is a independent variable, okay. So in this example with the gas, the amount of gas you put in can be whatever he wants. You could put in one gallon, you can put two, you could put in three. It doesn't really matter. You can put in however much your tank needs.
The second component is the dependent variable. And what that means is, in the gas case, this would be the cost, okay? You put in one gallon, you expect to pay a certain amount. The amount of gas you put in dictates the amount of money you pay, okay. It doesn't go the other way.
In general, when we deal with points on a line and all that stuff, we're dealing with dealing with x and y, x is our independent variable, y is our dependent, okay? x can be whatever it wants, let's write this a little bit bigger so you can make sure you can see it. x can be whatever it wants, y depends on that, okay? And to be a function that has to let be one of these dependent variables for each of those independent variables.

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