Well, you can have maps from \(\displaystyle\mathbb{R}^{{1}}\) to \(\displaystyle\mathbb{R}^{{2}}\); viceversa, and from \(\displaystyle\mathbb{R}^{{1}}\) to \(\displaystyle\mathbb{R}^{{1}}\). The graph of a function \(\displaystyle{f{{\left({x}\right)}}}:{X}\rightarrow{Y}\) is the set \(\displaystyle{\left\lbrace{\left({x},{y}\right)}:{y}={f{{\left({x}\right)}}}\right\rbrace}\) for some x, which is a subset of the Cartesian product \(\displaystyle{X}\times{Y}\) . Then you need \(\displaystyle{X}\times{Y}\) to be of dimension 3 or lower, or you would need 4 dimensions to do the graphing.