# Functions and equations

## Main Question or Discussion Point

Hey guys, I'm back.

I'm taking pre-calc this year as a sophomore in high school and had a few simple questions about functions.

1. What is the difference between a function and an equation?

2. Why do people sometimes write "y = f(x)"? What is "y = f(x)" saying?

3. Can I use "y" and "f(x)" interchangeably?

Mark44
Mentor
Hey guys, I'm back.

I'm taking pre-calc this year as a sophomore in high school and had a few simple questions about functions.

1. What is the difference between a function and an equation?
A function is a rule that pairs input values from some set to output values in another (possibly the same) set. An equation is a statement about the equality of two expressions.
2. Why do people sometimes write "y = f(x)"? What is "y = f(x)" saying?
This equation says that the y value is produced by some x value. For example, if f(x) = x2 + 3, f(1) = 12 + 3 = 4. If you sketch a graph of this function, one of the points on the graph will be (1, 4).
3. Can I use "y" and "f(x)" interchangeably?
You might run into trouble if you're working with two or more functions, f and g. For a given x value, each function will produce its own y value associated with that x value. If you're only working with one function, there shouldn't be a problem.

Thanks for replying! Now Im still a little confused with the second question. Does f(x) mean something different in y = f(x), or is it still talking about a function? Im reading it like some value y is equal to some function f(x), which doesnt make much sense to me.

HallsofIvy
Homework Helper
Strictly speaking "f" is a function while f(x) refers to the value of f for a given x.

The number, y, is equal to the number f(x).

If f is the "squaring function", f()= ()^2, then f(2)= 4.

A function is a combination of different kinds of equations. Whereas an equation is a combination of different variables and different symbols. A function can have multiple values according to the changing of variable values.

HallsofIvy
Homework Helper
A function is a combination of different kinds of equations. Whereas an equation is a combination of different variables and different symbols. A function can have multiple values according to the changing of variable values.
This is completely wrong. A function does not have to involve any "equations" at all. Mark44 gave a perfectly valid definition of "function" in his first post.

Mark44
Mentor
A function is a combination of different kinds of equations.
???
What do you mean by "different kinds of equations"? Two kinds that come to mind are true equations (e.g., 2(a + b) = 2a + 2b) and false equations (e.g. 1 = 0).

Also, how are these different kinds of equations combined?
Whereas an equation is a combination of different variables and different symbols.
So by your definition, would this be an equation?
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A function can have multiple values according to the changing of variable values.

A function is a term that is used when an input will only result in a single output.
For example:

y = x2 is a function because no two inputs will result in the same output.

the inverse, however, is not a function.

y = $$\pm$$$$\sqrt{x}$$ is not a function because a single input will result in multiple outputs. In this example, if we were to make x (the input) equal 16, then the output would be both 4 & -4.

A function is a term that is used when an input will only result in a single output.
For example:

y = x2 is a function because no two inputs will result in the same output.
Ummm... what is the output if I give 2 as the input? Then what is the output if I give -2 as the input?

If I'm not mistaken, those two inputs both result in the same output.

Mark44
Mentor
For example:
y = x2 is a function because no two inputs will result in the same output.
I think what you meant to say was "no input will result in more than one output."

I think what you meant to say was "no input will result in more than one output."
That sounds better, haha.

Hurkyl
Staff Emeritus
Gold Member
y = x2 is a function because no two inputs will result in the same output.
There's an input and an output there?

The equation "y = f(x)" is an assertion that the variable "y" and the expression "f(x)" are equal.

What such an assertion "means" depend upon what precisely you mean by "y" and "x". (and "f")

e.g. if the variables "x" and "y" are being used in a way such that "(x,y)" represents an indeterminate point of the coordinate plane, then "y = f(x)" is a predicate on points -- for any particular point, "y = f(x)" evaluates either to true or to false. Such predicates have many uses -- e.g. to specify the hypothesis of an interesting problem, or to name an interesting subset of the plane.

Sometimes, rather than letting all variables be independent indeterminates, one may decide to allow functional dependence. "y = f(x)" would then be an expression of that functional dependence. In such a context, rather than our mathematical language saying "x and y are variables denoting real numbers", we have decided it should say "x and y are variables denoting real numbers satisfying the functional relationship y = f(x)". e.g. if we specialize to the hypothesis that x=2, we must also specialize to the hypothesis that y=f(2).

Sometimes, would take the expression "y = f(x)" as defining a bit of short-hand, that anywhere we write the letter "y", we should instead replace it with the symbols "f(x)".

Often, several conventions are used simultaneously. Sometimes you even see the abomination "y = y(x)". Fortunately, for many purposes, the confusion isn't especially harmful.

Alas, you probably won't get a precise idea of all of these things without studying abstract algebra, differential geometry, or a categorical logic. (Others feel free to add other places where it might be encountered)

chiro
Another point to remember is that "x" of f(x) may not to be a single number (natural, rational, real etc) but it may be a mathematical structure like a vector or a matrix or something else that has a more complicated structure other than a single number.

There's an input and an output there?

The equation "y = f(x)" is an assertion that the variable "y" and the expression "f(x)" are equal.

What such an assertion "means" depend upon what precisely you mean by "y" and "x". (and "f")

e.g. if the variables "x" and "y" are being used in a way such that "(x,y)" represents an indeterminate point of the coordinate plane, then "y = f(x)" is a predicate on points -- for any particular point, "y = f(x)" evaluates either to true or to false. Such predicates have many uses -- e.g. to specify the hypothesis of an interesting problem, or to name an interesting subset of the plane.

Sometimes, rather than letting all variables be independent indeterminates, one may decide to allow functional dependence. "y = f(x)" would then be an expression of that functional dependence. In such a context, rather than our mathematical language saying "x and y are variables denoting real numbers", we have decided it should say "x and y are variables denoting real numbers satisfying the functional relationship y = f(x)". e.g. if we specialize to the hypothesis that x=2, we must also specialize to the hypothesis that y=f(2).

Sometimes, would take the expression "y = f(x)" as defining a bit of short-hand, that anywhere we write the letter "y", we should instead replace it with the symbols "f(x)".

Often, several conventions are used simultaneously. Sometimes you even see the abomination "y = y(x)". Fortunately, for many purposes, the confusion isn't especially harmful.

Alas, you probably won't get a precise idea of all of these things without studying abstract algebra, differential geometry, or a categorical logic. (Others feel free to add other places where it might be encountered)
We get it ... you're really smart! Whatever.

I'm pretty sure we all know what macbowes was saying when using the terms "input" and "output."

We have a function f(x). When we enter some value of x into the function (x is an input to the function), it results in a specific value (or values) as a result (the output).

So by input, he means variable(s) and by output, he means result(s).

HallsofIvy
Homework Helper
We get it ... you're really smart! Whatever.
And you believe that you are not? Or are you simply being insulting?

I'm pretty sure we all know what macbowes was saying when using the terms "input" and "output."
I'm not. "Input" and "output" are NOT standard mathematical terms while "function" and "equation" are.

We have a function f(x). When we enter some value of x into the function (x is an input to the function), it results in a specific value (or values) as a result (the output).

So by input, he means variable(s) and by output, he means result(s).
In other words, you do not know what "variable" means. The "result" of a function (another undefined word) is as much a variable as the "input".

Well, I would not assert that Hurkyl is smarter than you are. But it is clear from this that he knows more about mathematics than you do and is better at explaining it.

... are you simply being insulting?
Not any more than Hurkyl is being toward macbowes. In my opinion, he was belittling macbowes.

"Input" and "output" are NOT standard mathematical terms
While I completely agree with you on this point, I contend that everyone knows what macbowes meant in this case;

You put a value into the function, you get a value out. You enter an input, you get an output.

Besides, is this really constructive to helping the OP with this particular problem?
I thought we were all here to help each other. If someone is having a hard time understanding something, why can't we use "less-than-formal" terminology?

If I'm describing to you that I have heartburn, is it really necessary that I tell you that my hiatal hernia is acting up. No.

When a child asks how you make a car go faster, you tell him that you press down on the gas pedal. Never mind that it may actually be a diesel engine. Or for that matter that pressing the "gas pedal" is not what really what makes the motor run faster; it's the fact that you allow more fuel and air to flow into the engine, etc. As far as the child is concerned, the gas pedal is an acceptable answer.

I think you get the idea.

...it is clear from this that he knows more about mathematics than you do...
Really? It's clear from this? Just because I don't go into a dissertation on mathematics when trying to help out someone, that means it's clear that Hurkyl knows more about mathematics than me?

Is this supposed to be an insult?

The point is that we don't need to go into such great detail when helping someone who is obviously at a lower level than you.