What is the proper definition of a "function" and how does this relate to an "equation"?

Question

Dictionary.com gives several different definitions for "function" One definition of function says that it is a one to one mapping in which the function is uniquely defined for each element of the domain. This says to me that f(x) = square root of x would NOT be a function.

But, another definition says that it is "a mathematical relation such that each element of one set is associated with at least one element of another set" This definition means f(x) = square root(x) could be a function.

So I'm wondering which definition belongs in what contexts? And what is the relationship between an equation and a function? My understanding is that you might or might not be able to solve an equation for one variable. If you can solve it for one variable would that then be a function that you have?

Answer 1

Right, 2y-6=2x is an equation, but not a function. y=x+3, is a function where y is a function of x. It can also be written f(x)=x+3.

Answer 2

How in the world did you figure out that the square root is not a function.

f(x)=sqrt(x) is most definitely a function. You are thinking of f(x)=plus OR minus sqrt(x) which is not a function.

So here is a definition, the easiest way I can put it, a function is a relationship which gives you back no more than one value for f(x). Now think about this carefully. I chose the words "no more than one" to also imply that no answer for f(x) is also good.

So one x gives you one (or none) f(x), then f is a function.
But if one x gives you two different f(x), then f is not a function.

So now f(x)=sqrt(x) is a function and g(x) = plus or minus sqrt(x) is not a function.

For example, f(25) = 5 only so f is good.
g(25) = 5, -5 which is a no no.

That is where the vertical line test comes in. Remember, if the line crosses nothing, it is still okay. If the line crosses the graphof a function at one point, it is still okay. As soon as it touches two or more points on the graph at the same time, the relationship is not a function.

Answer 3

A 'relation' between the elements of a set A and a set B is called a 'function' ifit has the property that at most one element of B is related to an element of A.

A is usually called the 'domain' of the function and B is called the 'range' of the function.

It is important to notice that A and B are sets and, in general, do not have to even be the same *kinds* of things.

After that, such adjectives as 'injective', '1 to 1', 'linear', and so on are added to define properties and characteristics of the function.


Doug

Answer 4

A function has an input and an output. x is typically the input. The output is whatever is after the equation sign, i.e f(x)=x^2. Input is x. Output is x^2. There are infinite possible values in this case.
An equation gives that the two sides of the equation are equivalent, i.e. 2x=6. 2x and 6 are equivalent.

Answer 5

Proper definition of function:

It is a mathematical concept that describes a mapping of elements in a given domain to a given range.

The most common type of mapping is a linear transformation.

Answer 6

the first definition is correct according to what i learned in matriculation. one object can only have one image for a function.

but, i'm not sure about square root of x would not be a function because when you want to define f(x)=surd x, we'll take the positive only. i can't remember why.

Answer 7

Happiness is a state that exists only for a moment

Answer 8

The function f(x) is part of an equation... f(x) = (whatever something is)...

Answer 9

compute single or multiple integrals, solve systems of nonlinear equations, find maxima and minima, compute derivatives, find roots, plot graphs.