 One of the most important concepts in higher mathematics is that of a function. And this emerges as follows. In many situations, we have a relationship between several variables. For example, there is a relationship between the area a, height h, and width w of a rectangle. There is a relationship between the x and y coordinates of a circle centered at the origin with radius r. There is a relationship between the amount of money a politician receives from their corporate sponsors and the votes they cast. Sometimes we can write an explicit formula for the relationship. So for the area relationship, we can write this as a equals h w. For the relationship between the x and y coordinates of a circle, we can write that as x squared plus y squared equals r squared. And the relationship between the amount of money a politician gets from their corporate sponsors and how they vote, the relationship is, well, maybe we can't write a formula for that. And maybe that's part of the problem. Now we say that these quantities like area, length, width, corporate money are all variable, but we may be interested in the value of one specific variable based on the values of the others. So we might want to know the area of the rectangle based on the length of its sides. Or we might want to know the x coordinate of a point based on its y coordinate. And we're almost certainly interested in knowing the vote of a politician based on his corporate sponsors. The variable we're interested in is called the output variable and the others are the input variables. And here's an important idea. Relationships are most useful when they produce one and only one output for any given input values. And this leads to the idea of a function. A function is a relationship between variables that has one output for any given input. In that case, we say the output variable is a function of the input variables. So for example, is the area of a rectangle a function of the length and width of a rectangle? Definitions are the whole of mathematics, so let's pull in our definition of a function. So we want to check to see if there's only one output for any given input, which means we have to identify what the output and input are. Now the wording function of the length and width suggests that the length and width of the rectangle are the input variables. Meanwhile, the area of the rectangle seems to be the output variable. So we want to know if there is one and only one output, the area, for any given inputs, the length and width. And so we might make the following claim. Given any specific values of the length and width as inputs, there is one and only one possible area. So area is a function of length and width. How about, is the length of a rectangle a function of its area? So again, the wording suggests that the input variable is the area while the output is the length. So again, pulling in our definition of a function, we want to know if there is one and only one possible length of a rectangle for a given area. And the answer to this question is no. Given any specific area for a rectangle, there could be more than one length that produces the given area. So length is not a function of area. Now we might actually have an algebraic relationship between the variables, for example, y equals x squared. Is y a function of x, or is x a function of y? So if x is our input, there's only one possible value of x squared, so there's only one possible value of y. y is a function of x. If y is our input value, then x is a square root of y. But there could be more than one square root, so x is not a function of y. Or we can have the following. y is the square root of x. y is, remember how you speak influences how you think, principle square root of x. So definitions are the whole of mathematics. All else is commentary. y is the square root of x. Well, what's our definition of square root? So remember, if b squared equals a, we say that b is a square root of a. But this means that if b is a square root of a, so is negative b. So the square root of x is not a function. On the other hand, remember that this symbol refers to the principle square root of a, which is the non-negative number whose square is a. And so since principle square root of a is the non-negative number whose square is a, and there can be only one, then principle square root of x is a function.