Are functions a subset of equations?

[V0ODO0CH1LD]

I read a few posts about the matter but they seemed to contradict each other. So what are the differences between a function and an equation? Furthermore; what explains the fact the some of them are graphed the same even though they are not the same?

I get that functions are defined as mappings from one set to another with the restriction that one input must only map to one output. But then what are equations?

One of the explanations I read was that functions output a value for every input and equations only show the relationship between variables. But then the examples confused me because "f(x) = x" was said to be a function and "x - y = 0" was said to be an equation. But can't "x - y = 0" be viewed as the function of x and y such that any input outputs zero? I mean, there are such things as constant functions, right?

 

[Jakeus314]

I would say that x-y=0 can be written as a function. And given that relationship, x is a function of y and y is a function of x.

 

[gerben]

An equation is simply an expression in which you state that two "things" are equal, like the expression a=b in which you state that a is equal to b. In the equation f(x)=x, f(x) is notation for "the value to which x is mapped by the function f", so the expression simply says "the value that function f maps x to is equal to x".

 

[Boorglar]

I think a distinction would be the following:

1 = 1
1 = 2*7 - 13

are equations (they state the equality between the left side and the right side) but you cannot really interpret them as functions, since there is nothing mapped into something else.

the exponential function is a mapping from R to R which gives e^x as an output when given x as input.
So a function does not state the equality between two things, but rather "transforms" a given input in some way

To sum up, I see equations as more "static", while functions are "interactive", in some way...

 

[gerben]

I think a distinction would be the following:

1 = 1
1 = 2*7 - 13

are equations (they state the equality between the left side and the right side) but you cannot really interpret them as functions, since there is nothing mapped into something else.

the exponential function is a mapping from R to R which gives e^x as an output when given x as input.
So a function does not state the equality between two things, but rather "transforms" a given input in some way

To sum up, I see equations as more "static", while functions are "interactive", in some way...

Or maps each element in the input set to an element in the output set, a function is just simply that mapping.

When you mention f(x) you are referring to an element in the output set; x is an element in the input set and f(x) is the element in the output set to which the function f maps x.

 

[arildno]

Short answer:
f(x)=x is an IDENTITY, not an equation, holding true for every choice of x.
It simply defines the function VALUES, and is not the function itself.

 

[Number Nine]

But then what are equations?

Equality is an equivalence relation defined on a set. An equation is then just a statement that two members of a set are equal. A function is something else entirely, being a mapping between sets.

 

[gerben]

Short answer:
f(x)=x is an IDENTITY, not an equation, holding true for every choice of x.
It simply defines the function VALUES, and is not the function itself.

The statement f(x)=x is an equation. It is stating that f(x) is equal to x. It is a statement of equality between f(x) and x and thus an equation.

(the function f defined by this equation is the identity function, often called the identity because it is a mapping of every element to itself)

 

[V0ODO0CH1LD]

Does that mean that something like "f(x) = mx + b" is a function being expressed as an equation?

 

[jbriggs444]

Does that mean that something like "f(x) = mx + b" is a function being expressed as an equation?

I would say that it is a function being specified using an equation.

To make the distinction a bit more concrete...

x^2 + y^2 = 1 is an equation.

The set of ordered pairs (x,y) that satisfies that equation is a relation.

The set of ordered pairs (x,y) that satisfies that equation and has y >= 0 is a relation that is the graph of a function. [By some definitions, such a relation _is_ a function].

f(x) = sqrt(1-x^2) is a formulaic specification of that function.