Why is "Implicit Function" a Misleading Translation?
Definition
The distinction between a positive and a negative function merely depends on how each is represented. Although it’s a somewhat unfamiliar expression in mathematics, the differentiation depends on how ‘independent variables’ and ‘dependent variables’ are represented. Simply put, it involves setting the independent variable as $x$ and the dependent variable as the changing $y$ and observing their forms.
Examples
For example, in the case of $y = x^2 + 1$, by setting it as $f(x) = x^2 + 1$, $y = f(x)$ is obtained, and it can be said to be a positive function. On the other hand, the case of $x^2 - y + 1 = 0$ represents the same equation as a negative function expression.
Another example is the negative function expression $x^2 + y^2 -1 = 0$. It doesn’t matter whether it’s an expression of $y$ for $x$ or $x$ for $y$, but upon rearranging, it becomes something like $x = \pm \sqrt{1- y^2}$, which cannot be a positive ‘function’. The above expression is nothing but a combination of the positive function expressions $x = + \sqrt{1- y^2}$ and $x = - \sqrt{1- y^2}$.
Inefficient Naming
As can be seen in the examples, a positive function can be expressed as a negative function, but the reverse may not be possible. Therefore, the term ‘positive function’ is only used when emphasizing the contrast with negative functions and is otherwise not used.
Now, to talk a bit about the translations, the English terms for positive functions are Explicit Function and for negative functions Implicit Function. The words explicit and implicit translate to ‘clear’ and ‘implied’, respectively. Anyone can see that these words were chosen as antonyms and have seeped into the definition of mathematics, with the contrasting Chinese characters being Ming明 and An暗. The issue is that while Yang阳 and Yin阴 might be appropriate for such opposition, in mathematics, these are already represented by Positive and Negative. Logically, such expressions could cause significant confusion for high school students who have a decent sense of language.
Not only is it inappropriate in terms of nuance, but it is also the same from a practical perspective. Frankly, it’s inefficient to have given the important characters ‘positive’ and ’negative’ for the not-so-frequently used terms, positive and negative functions. Even now, many textbooks use frustrating expressions like ‘a function value that is not positive $f$’, which certainly needs improvement. Those who have studied in English and tried to express it in Korean would empathize with how much a short and concise expression like non-positive is needed.
I dare to suggest that the current ‘positive function’ and ’negative function’ be translated to ‘Ming function’ and ‘An function’ respectively, which seems more appropriate from various aspects. While I cannot claim this proposal to be the best, I am confident that it is better than the current terms.