A function maps all elements of its domain (each) to one element of its range. The vertical line test is a graphical way to show that for a graph of a function mapping one variable to another in the Cartesian plane. But there is no reason why a function can't have any of the following forms: Map a real number, x to a complex number x+ix² or map a pair of points satisfying the equation x+y = 3 to x*√|y|. It is arbitrary whether you choose to think of the result as existing "in the same graphical space" as the other two. In my first example it might be obvious that the result can't be "in" the same (one dimensional) space as the values of x. You can define a function of 3 independent variables onto a 1537 dimensional space (to give a random example). Mostly our brains get confused above two or three independent variables, so most of your math will be restricted to low dimensions, and usually (but not always) the domain and range are both subsets of the same space (like 2D, 3D or 4D). A line will only intersect a function that extends from -∞ to +∞ IF you restrict the function to "live" in 2D. The same function in 3D could easily avoid intersecting any given line at all. The vertical line test fails in three ways I can think of: first I can Graph y=x² with the x-axis vertical. Drawing a vertical line doesn't test whether y is a function of x in that case. I can define a function that = 0 when x < 1 and =7 when x >= 1. There will be points arbitrarily close to x=1 but infinitessimally less where the function = 0 so any "physical" line drawn at x=1 will also overlap those points (since any 'physical' line has a finite width). And third a line only works, as discussed above, in Euclidean 2D spaces, most functons aren't found there. The test is a good way for beginners to gain an intuitive 'feel' for functions, and gives some practice at identifying those relationships that are and those that are not functions. Oh, btw any continuous curve can be divided (cut up) into functions. Using a vertical line allows you to identify where to make the cuts...

A function maps all elements of its domain to one element of its range. The vertical line test is a graphical way to show that for a graph of a function mapping one variable to another in the Cartesian plane. There is no reason why a function can't have any of the following forms: Map a real number, x to a complex number x+ix² or map a pair of points satisfying the equation x+y = 3 to x*√|y|. It is arbitrary whether you choose to think of the result as existing "in the same graphical space" as the other two. 

In my first example it might be obvious that the result can't be "in" the same (one dimensional) space as the values of x. You can define a function of 3 independent variables onto a 1537 dimensional space (to give a random example). Mostly our brains get confused above two or three independent variables, so most of your math will be restricted to low dimensions, and usually (but not always) the domain and range are both subsets of the same space (like 2D, 3D or 4D). A line will only intersect a function that extends from -∞ to +∞ IF you restrict the function to "live" in 2D. The same function in 3D could easily avoid intersecting any given line at all. The vertical line test fails in three ways I can think of: first I can Graph y=x² with the x-axis vertical. Drawing a vertical line doesn't test whether y is a function of x in that case. I can define a function that = 0 when $x < 1$ and =7 when $x ≥ 1$. There will be points arbitrarily close to x=1 but infinitesimally less where the function = 0 so any "physical" line drawn at x=1 will also overlap those points (since any 'physical' line has a finite width). And third a line only works, as discussed above, in Euclidean 2D spaces, most functions aren't found there. The test is a good way for beginners to gain an intuitive 'feel' for functions, and gives some practice at identifying those relationships that are and those that are not functions. 

Oh, btw any continuous curve can be divided (cut up) into functions. Using a vertical line allows you to identify where to make the cuts...