For certain types of functions I have been using the term 'general form' as a way of stating that some function of some variable(s) is true for more than one value of the variable (usually an infinite number). Here is a simple example, 1=cos(2∏x) There have been (at times) where the use of this term has caused confusion so it would seem this may not be the best terminology, any suggestions would be appreciated.
You lost me. You can talk about equations or inequalities being true, but functions generally evaluate to numbers. The equation above is true for specific values of x; namely, when x is an integer. I don't see how 'general form' applies here.
That is surprising considering the function (for this example) evaluates to a number and is a representation where this statement is 'true', as in if we were instead looking for a 'solution' to the variable then one could understand 'being lost'. In light of all this it would seem I should not be using the term 'function' and instead replace it with 'equation' so as to avoid confusion for cases of this type. Thanks for the insight! Excellent point, I should have added ##\{x\in \mathbb{N}\}## to the example. Considering this correction (and my apologies for having not added this important detail) would stating the function, 1=sin(2∏x) ##\{x\in \mathbb{N}\}## is a 'general form' be correct terminology? If not then what would be?
In the equation of your example, sin(##2\pi x##) is a function that we can evaluate for arbitrary values of x. When you add "= 1" you are constraining the possible values of x that make the equation a true statement. Yes. I don't see why. All you're doing is explicitly stating the solution set for the equation. No, it's not a general form. Again, all you have is an equation and its solution set. An example of a general form is the equation Ax + By = C, which is the general equation of a straight line in two dimensions. A, B, and C are parameters here, and x and y are the variables. This is a general form of a line equation because every line in the plane is described by specific choices for the three parameters.
I understand, Hallsofivy had added similar insight while giving information about family of functions. So when speaking about 'general form' this term is used to refer to a change in parameters for which each results in an entirely new function. Is the term 'general form' restricted to apply when a family of functions represents every possibility, e.g. (Ax=D), (Ax+By=D), (Ax+By+Cz=D), etc. for which ##\mathbb{R^n}## the functions are in? Finally, what is the correct term to use to describe function like the example that was laid out initially?
Or general equation. The first equation above indicates a point on a line ; the second, a line in the plane (R^{2}); the third, a plane in space (R^{3}). I don't think there is a special term. It was just an equation involving a trig function.
A function f(x) whose range is the real numbers doesn't have the range { "true", "false"} so you you shouldn't speak of the function as being "true". An equation can be regarded as a statement that is a function of variables. From that point of view, an equation can be regarded as a function that has the range {"true", "false"}, so it isn't completely improper to speak of an equation as being "true" for certain values of the variables. However, a better terminology is to say that an equation "is satisfied" by certain values of the variables. If you say "This equation is true" or "This equation is false" it might be interpreted to mean that the equation does or does not correctly describe some paricular law of physics or mathemetics. An equation f(x) = c for an arbitrary constant c has at most 1 solution when f is a 1 to 1 mapping from the real numbers to the real numbers. If you want to say that there exist particular values of c for which f(x) has more than one solution, you could say that f(x) is "many to 1" or "not 1 to 1". If you want to pick only one particular value of c, such as in the equation f(x) = 1 then I don't know of any concise terminology that says the equation has multiple solutions. So just say that the equation f(x) = 1 has multiple solutions. Perhaps you aren't thinking about the function f(x) itself. You might be thinking that the equation f(x) = c implicitly defines a function x = g(c) that "solves the equation for x". Is that what you want to know about?
To me, it seems "a function of some variable is true for more that one value of the variable" would be a "non-injective" function or "not a one-to-one" function. My interpretation is that you are saying that there exist at least two inputs that give the same output value. The function is also periodic: that seems a more important term to use for trigonometric functions of this type. The "general form" is just a structuring of an equation so that you can easily classify the graphical and numerical properties. In other words, if I look at ##y = x^2 -6x + 5##, I can tell you a lot of things without actually graphing or making tables of values.