I'm currently in Pre-Calculus (High School), and I have a relatively simple question that crossed my mind earlier today. If I were to take a graph and draw a random line of any finite length, in which no two points along this line had the same $x$ coordinate, would there be some function that could represent this line? If so, is there a way we can prove this?
Functions – Is There a Function for Every Possible Line?
functionsgraphing-functions
Related Solutions
Draw a line through the origin, $y=mx$, and mark the point $(1,m)$. Rotate the drawing by a quarter turn and observe that the point is now at $(-m,1)$.
Old slope: $m$, new slope: $-\frac1m$.
I would say prime notation is a bit out of its depth here. If you are taking the derivative of a multivariate function $f(x,y)$, you would need to specify the variable with respect to which you are differentiating.
For example, letting $f(x,y)=x+y$: $$\frac{\partial}{\partial x}f(x,y)=y+1$$ $$\frac{\partial}{\partial y}f(x,y)=x+1$$ using Leibniz notation. If you haven't seen this notation, the choice of $x$ or $y$ in the denominator of the $\frac{\partial}{\partial x}$ tells the reader that you are not interested in the other variable. I have never seen the notation $f'(x,y)$ used, and I don't think it would be considered acceptable by most authors. Hope this helps!
P.S. I'm in my final year of high school, so hopefully this isn't hard to understand :)
Best Answer
As an aside to supplement the answers you have already gotten, I would like to point out that a function doesn't need to have a formula to be considered a function. This is because a function is just a relation between two sets (the domain and range), with the property that you get only a single output for each input (this is the rule that there can only be one $y$ value for each $x$ value).
For instance, I can define a function $f$ from the set $\{0,1,2\}$ to the set $\{5,10,15\}$ by giving the rules
$$f(0)=10 \\f(1)=5 \\f(2)=15$$
It might be difficult or impossible to find a formula for $f$ in which you can plug in $0$, $1$, or $2$ and get the correct answer as a result, yet $f$ is a function since each input gives me a unique result.
By the same token, you can draw a straight line, or a wiggly line, a line with holes in it, or just a bunch of points, etc, and as long as you have ensured that there is no point on the graph where a single $x$ value gives more than one $y$ value you have essentially defined a function. This is true whether you can find a formula for it or not.