If you think about graphing an equation like f(x) = x; you think about a line through the origin in two dimensional space, where the horizontal axis represents the domain and the vertical the image. How can you get the input and the output of a transformation in the same picture? In the previous case neither the input nor the output even exist in ℝ^2. If you take that to higher dimensions it becomes even weirder. How can a transformation from ℝ^n to ℝ^m be represented in ℝ^(n+m). I get how it works, we've been introduced to the concept in middle school. I'm just wondering if there is some elementary reason or if it's just convention. And also, who decided that the constant output of a function like g(x, y) = x - y = 0 should be omitted from its graph? Why is it that in this case we're allowed to map only all the possible inputs to the transformation and and leave out the fact that a third component z exists but is just a constant. By the f(x) = x graphing convention shouldn't the graph of x - y = 0 be a line through the origin in three dimensional space that stays in the xy-plane?