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.
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.
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
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".
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.
B.S. Earth Sciences; M.S. Geophysics
Oklahoma State Univ. Inst. of Technology
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Update: June 2012