High School Equations vs. Functions Quadratic and Cubic?

[shintashi]

So if i take the rules that a straight vertical line drawn through the function with more than one intersection implies it is not a function, to mean that the quadratic equation for a circle is not a function.

Furthermore, it also implies a cubic equation, such as x^3 can be a function, because the solutions will not have both a positive and negative. And if I am to generalize, it can be said that all even powers, quartic functions, etc., i.e., if the power is such that the equation has x^2, x^4, x^6, x^8, etc., then it is NOT a function,

but lacking any of those if an equation has odd powers, X^1, X^3, X^5, X^7, etc., any of these or combinations would generally be a function? So X^4+X^2 would be NOT a function, while X^7+X^3+nX^1 etc. would be, and if we mix the odd and even, the even powered variables automatically cause the whole equation to fall outside of a function, thus x^5+X^4+X^3 is canceled as a function because of the X^4.

is there anything else i should look for in identifying the differences?

 

[mfb]

So if i take the rules that a straight vertical line drawn through the function with more than one intersection implies it is not a function, to mean that the quadratic equation for a circle is not a function.

There are more general functions where you can have a circle, but on a high school level: Yes.

And if I am to generalize, it can be said that all even powers, quartic functions, etc., i.e., if the power is such that the equation has x^2, x^4, x^6, x^8, etc., then it is NOT a function,

Huh? A vertical line will intersect f(x)=x² exactly one, never twice. It is a function.

and if we mix the odd and even, the even powered variables automatically cause the whole equation to fall outside of a function, thus x^5+X^4+X^3 is canceled as a function because of the X^4.

That doesn't make any sense.

 

[shintashi]

ah, i was thinking with a rotation of 90 degrees with my x and y reversed (in blender Z is vertical and X and Y are subjective)
f(y) doesn't work with the above formulas if a function is canceled by a vertical line.
for example:
x = y^3, x = y^1, x = y^3+y^5, x = y^3+y^7 all of these produce graphs as functions of y,
but
x = y^4, x = y^2, x = y^4+y^6, x = y^2+y^6 etc. are all NON functions because of the vertical rule...

or am i missing something? does f(y) exist?
does the vertical rule apply to functions of y? or do functions of y use a horizontal rule?

 

[shintashi]

https://ibb.co/nOQSES

here's a comparison graph of what my brain was thinking but had turned 90 degrees. So do functions of y exist and do the rules for non functions remain vertical lines or do they switch to horizontal lines?

 

[Mark44]

ah, i was thinking with a rotation of 90 degrees with my x and y reversed (in blender Z is vertical and X and Y are subjective)
f(y) doesn't work with the above formulas if a function is canceled by a vertical line.
for example:
x = y^3, x = y^1, x = y^3+y^5, x = y^3+y^7 all of these produce graphs as functions of y,
but
x = y^4, x = y^2, x = y^4+y^6, x = y^2+y^6 etc. are all NON functions because of the vertical rule...

All of the above are functions, with x being a function of y. In all of these equations if we assume that x is the dependent variable and y is the independent variable, these would usually be graphed with the vertical axis being used for x and the horizontal axis being used for y. With this assumption a vertical line would intersect (not cancel) the graph once.

If you solve each of the 2nd set of equations for y, so that y is in terms of x, then none of these (in the 2nd set) is a function. For example, if you solve x = y^2 for y, you get $y = \pm \sqrt x$. Each x value, with x > 0, pairs with two y values

or am i missing something? does f(y) exist?

In all your equations, both sets, you have x as a function of y, or x = f(y). In that regard, they are all functions. It's only when you solve each of the equations in the 2nd set for y in terms of x, that you don't also have y as a function of x.

does the vertical rule apply to functions of y? or do functions of y use a horizontal rule?

The vertical rule applies when the dependent variable is on the vertical axis and the independent variable is on the horizontal axis.

 

[shintashi]

ok, so, to clarify

vertical rule applies to
y = f(x) = x^2+x^4+x^6...
y = f(x) = x^1+x^3+x^5...
or a combination of any of these, and because they are not subject to a horizontal rule, they are still functions...

But,

horizontal rule applies to
x = f(y) = y^2+y^4+y^6...
x = f(y) = y^1+y^3+y^5...
or a combination of any of these, and because they are not subject to a vertical rule, they are still functions...

and ultimately, whatever axis "f(n)" translates to, that is the rule it follows:
i.e., if f(n) is used to plot y, or equals y, it has a vertical rule,
and if f(n) is used to plot x, or equals x, it has a horizontal rule.

Something i don't understand though about dependent and independent variables:
in x^2 + y^2 = 1
mathwords.com says x is the independent variable, and y is the dependent variable.
Why? Why not the other way around, or why not just 1 as the dependent variable?

 

[mfb]

I don't think these "vertical rules" and "horizontal rules" are very useful.

A function must have a unique value for every argument ("input"). $f(y)=y^2+7y^5$ has a unique value of f(y) for every y. It is the same function as $f(x)=x^2+7x^5$ or $f(\alpha)=\alpha^2+7\alpha^5$. How you plot this function is a different question.

in x^2 + y^2 = 1
mathwords.com says x is the independent variable, and y is the dependent variable.
Why? Why not the other way around, or why not just 1 as the dependent variable?

It is arbitrary what you consider independent and what you consider dependent. The two variables depend on each other, but the situation is clearly symmetric.