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Name: Bill
Status: student
Grade: 9-12
Location: GA
Country: USA
Date: N/A 


Question:
Hi, Thank you for looking at my question. I am a sophomore, and I have a quick question about functions. How do you know if something is a function. For example, I know that y=x^2 is a function but what about other things. Is a scatter plot a function? It can pass the vertical line test, so is it a function? Is a normal curve a function? What I am trying to do is differentiate between two things. What I am thinking is that everything plotted on a coordinate plane is a function, but I do not think that is true, for some things are just data points right? I am a bit confused.



Replies:
Bill,

If i understand your question correctly, a function is simply an equation in which the results are dependent upon another variable or variables. So y=x^2 is y as a function of x, or y(x). Another example: The hypotenuse can have a maximum number of two variables; say a, and b. If neither are defined as constant, then the hypotenuse is a function of a and b, or c(a,b)=sqrt (a^2 + b^2). So any equation that has dependent variables, is considered an equation as a function of a dependent variable or variables.

Hope that helps.

-Alex Viray


Bill

A function is a rule that assigns things, such as numbers, to other things. If each object in a set is assigned to one other thing, in the same set or another set, that assignment is a function. When the function assigns a number to another number and it is graphed on a coordinate plane, then the graph satisfies the vertical line test, as you noted. On the other hand, if you have a bunch of points on a coordinate plane, such as a data plot or scatter plot, that happens to satisfy the vertical line test, then that set of points can be used to define a function. The only thing that matters is that two or more things are not assigned to the same thing.

Data plots, probability distributions, such as the normal curve, are perfectly good functions if their graphs satisfy the vertical line test. The domains would be the set of first coordinates and the range the set of second coordinates.

Paul Beem


A mathematical function is a very general idea, so do not make it over-complicated. Think of the "independent variable" as one (or more) input variables. In your example, Y = X^2 that would be "X", but it could be as many as necessary for the given problem -- (X1,X2,X3, ...). The number of input variables could even be infinite, but let us just keep it simple and use a single one, X.

Think of "the function" f(X) as a set of instructions that tells you what to do with X. A good analogy is the function is a "black box" with a handle that you are going to crank. What "pops out" at the other end of the "black box" is the "dependent variable". In your example X^2. That's all there is to it. It's no more complicated than that. The "black box" may be simple or complicated, but that doesn't change the definition of "a function". The "function" is just that set of instructions.

Of course, there may be a different set of instructions for X1, X2, X3, ... but that does not change the definition of "a function".

Vince Calder


A function is a predictable, well behaved equation. If an equation is a function of (in) x, then there will be exactly the same result every time you put the same value of x into the equation. In other words, for every value of x there will be one and only one value of y. y = mx + b is a function.

Let us say you have a job after school and you are paid by the hour. You want your boss use an equation that gives you the same pay for the same amount of work each week. Think what would happen if they used an equation to calculate your pay that had two answers! Which answer do you think your boss would pick for your paycheck? You want your pay to be a function of the hours you worked. Pay = f(hours).

Not all equations are functions and not everything plotted on a set of axes is a function. x = y^2 is not a function because for x = 2 you could have y equal both 2 AND -2. So, when we find one of those well behaved equations we award it, if you will, the f badge as a sign of its dependability and good behavior. In addition we put the variable that defines the answers in the parentheses as an aid to those using it. For example, if we have the equation f(x) =Gxy that means the values of G and y stay the same and only the value of x changes. In some really complicated equations with lots of variables, this notation is a life saver.

In answer to your query on scatter plots. Yes they represent functions UNLESS there is a value of x (the x coordinate) that has two values of y connected (A repeated value of x that has a different values of y such as (2,3) and (2,7). Of course, you may go crazy trying to write the equation for the plot.

Hope this helps, but if you need more information write me directly as I am teaching Remedial Intermediate Algebra this semester and can send you a write up on functions.

Bob Avakian
Instructor
B.S. Earth Sciences; M.S. Geophysics
Oklahoma State Univ. Inst. of Technology



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