Why do we use squared quantities in equations and formulae?

[Mark44]

@sophiecentaur I really have a hard time understanding what you're trying to say. The fact that functions are single valued (by definition) is a critical property.

Strongly agree. In the example posted by @sophiecentaur, $x^2 + y^2 = 1$, y is most definitely NOT a function of x. The same is true for the other conic sections. For the circle equation, solving for y yields $y = \pm \sqrt{1 - x^2}$, so for each x, with $x \ne \pm 1$, there are two y values.

Afaics, it's important to qualify statements about the 'meaning' of Root x.

No. Although a positive real number has two square roots, the symbol $\sqrt x$ is taken by convention to mean the principal, or positive square root. Why else do you think there's a +/- symbol in the Quadratic Formula; i.e., the solutions to $ax^2 + bx + c = 0$ are given by $x = \frac{b^2 \pm \sqrt{b^2 - 4ac}}{2a}$.

I think you mean a continuously differentiable function. There are plenty of functions with first derivatives that aren't continuous. Triangular waves, cycloids and x² = y

No, he (@etotheipi) meant plain old functions. BTW, $x^2 = y$ is continuous and has derivatives of all orders. $\frac{dy}{dx} = 2x$. OTOH, if you meant $y^2 = x \Leftrightarrow y = \pm \sqrt x$, then y is not a function of x. (Each positive x value is paired with two y values.)

Perhaps the vocabulary has changed in 50 years. (Which is possible.)

No, it hasn't. From Calculus and Analytic Geometry, by Abraham Schwartz, 2nd Ed., published in 1967: (emphasis added by me)

Function; domain; range. We are given a set of numbers, which we shall call the domain D, and instructions for associating a number y with each number x of D. The set of all numbers y associated with numbers x of D shall be called the range R. The correspondence thus created between the sets D and R shall be called a function.

The relationship here is either a one-to-one relationship (e.g. y = f(x) = 2x) or many-to-one (e.g., y = g(x) = sin(x). By definition, a function cannot be one-to-many (e.g. $y^2 = x$ or equivalently $y = \pm \sqrt x$).
The First Edition appeared in 1960, which is 60 years ago. I'm positive the definition of a function didn't change between the two editions.

"By definition" is too strong. You are stating that the default actually exists.

No, "by definition" is not too strong. One definition of the absolute value is $|x| = \sqrt{x^2}$, for all real numbers x.

How can "by definition" be too strong? It just a convention about what a word means in a certain context, that is most widely accepted.

 

[sophiecentaur]

I do take all of your points and I clearly have mistaken how the word "function" is strictly defined. So the term is mi-used a lot? Perhaps and it wouldn't be the only commonly mis-used term.

 

[Mark44]

I do take all of your points and I clearly have mistaken how the word "function" is strictly defined. So the term is mi-used a lot? Perhaps and it wouldn't be the only commonly mis-used term.

Yes, it's misused a lot. A test that can be used to determine whether a graph represents a function is the vertical line test. If a vertical line is swept along the horizontal axis and never intersects more than one point, the graph is that of a function. This would show that the graph of a circle isn't that of a function, nor is the graph of the other conic sections, including, say, $y^2 = x$.
Also, we have had many threads over the years where people are confused about the meaning of $\sqrt x$, thinking that it represents two values.

 

[sophiecentaur]

Also, we have had many threads over the years where people are confused about the meaning of $\sqrt x$, thinking that it represents two values.

Ahh - and the 'quadratic solving formula' actually includes the term +/-√(4ac) to take care of that. It was hiding from me in plain sight!