Understanding the Absolute Value Formula: Is it a Rule or a Myth?

[mather]

hello!

is this a formula?

absolute value of y = absolute value of x

because I read a rule somewhere that you cannot have a formula where many x have the same y or something

thanks!

 

[stefan10]

You are most likely confusing the concepts of a formula and of a function. Formulas are best described as relationships of two or more variables. In mathematics they are derived using mathematical logic, which can incorporate function(s). Formulas can be represented as functions (either one function or a number of functions.)

Here is an example of a formula.

Volume of a sphere: V = (4/3)*∏*r³

Functions (simplified) on the other hand are any relationship between two or more variables which do not have multiple different outputs for a given input. For a function with output y and input x (or any relationship of two variables) there must only be one y (output) for a given x (input.) So it is the opposite of what you said. You can't have many y for a given x, if you wish to have a function. The above formula for the volume of a sphere can be written as a function.

V (r) = (4/3)*∏*r³

Notice that V or V(r) is dependent on which specific value you choose for r. We give this type of variable a name. It is the dependent variable. We also give the type of variable that we see with r, or x, or any other input variable a name, it is called the independent variable.

So using this information, can you find a value for x, in the equation you gave, that has two values of y? If not, it is a function. If you can, then it is not a function.

Now if you are concerned with whether or not it is a formula, the answer is yes, it is. It represents a relationship of two (or more) variables, and therefore it qualifies as a formula. The only equations that are not formulas are those with only one variable.

I hope this has clarified your understanding.

 

[mather]

sorry, I mean, is it a function?

 

[Mark44]

sorry, I mean, is it a function?

No.

The equation |y| = |x| is a relation. It is not a function, because each y value except 0 is paired with two x values. The graph of this equation is the same as the graphs of y = x and y = -x combined.

Edit: What I meant to write, but didn't, was that each x value except 0 is paired with two y values. The so-called vertical line test shows that the graph of |y| = |x| is not a function.

 

[stefan10]

Nevermind, I misread the post.

 

[skiller]

The equation |y| = |x| is a relation. It is not a function, because each y value except 0 is paired with two x values.

I think you have your x and y the wrong way round there.

 

[HallsofIvy]

Yes. (Which is very surprising for Mark44!) What he should have said is that |y|= |x| is not a function because each value of x except 0 is paired with two y values.

 

[BruceW]

it's not necessarily the wrong way around. Mark might have been talking about |y| = |x| as a map from y to x. In which case, it is not a function from y to x. This is essentially the same as saying |y| = |x| is a map from x to y, and in this case, it is not a function from x to y. So anyway, it is only convention that says x gets mapped to y. And this convention is not always used.

 

[Mark44]

No, to come clean, it was a mistake on my part. What I wrote says that the equation is not one-to-one, which isn't what I meant to say.