Is it true that for any series of numbers that when graphed make a continuous line with one and only one y value for each x value, that that series has a unique formula?
Yes, if we can agree on the term unique formula.
Does turning a curve, or a set of numbers which make up a curve, into a formula have a name? Is there a methodology I can study to learn how to do this?
No, that statement is not necessarily true. It is not always the case that a series of numbers, when graphed as a continuous line, has a unique formula. While some series of numbers may have a specific formula that can represent them, others may not have a simple or explicit formula at all.
In mathematics, a continuous line with one and only one y-value for each x-value is known as a function. Functions can be represented by formulas, but not all functions have a simple algebraic formula that can express every possible value in the series.
There are various types of functions that may not have explicit formulas:
1. Piecewise functions: These functions have different formulas for different intervals or ranges of x-values. For example, a function that is defined differently for positive and negative x-values.
2. Recursive functions: These functions are defined in terms of previous or subsequent terms in the series. Each term depends on the preceding term(s). These types of functions are commonly found in sequences or series.
3. Transcendental functions: These functions involve non-algebraic operations such as exponentials, logarithms, trigonometric functions (sine, cosine, etc.), or their inverses. Transcendental functions might require numerical methods or special techniques to approximate or represent them.
While some mathematical techniques, such as calculus or numerical analysis, can help approximate or analyze functions without explicit formulas, it is important to note that not all functions can be easily expressed using a single, closed-form formula.