Although the use of diagrams in arguments in analysis faces special dangers (as noted in 3.3), the use of diagrams to illustrate symbolically presented operations can be very helpful. Consider, for example, this pair of operations \(\{ f(x) + k, f(x + k) \}\). Grasping them and the difference between them can be aided by a visual illustration; similarly for the sets \(\{ f(x + k), f(x - k) \}\), \(\{ |f(x)|, f(|x|) \}\), \(\{ f(x)^{-1}, f^{-1}(x), f(x^{-1}) \}\). While generalization on the basis of a visual illustration is unreliable, we can use them as checks against calculation errors and overgeneralization. The same holds for properties. Consider for example, functions for which \(f(-x) = f(x)\), known as even functions, and functions for which \(f(-x) = -f(x)\), the odd functions: it can be helpful to have in mind the images of graphs of \(y = x^2\) and \(y = x^{3}\) as instances of evenness and oddness, to remind one that even functions are symmetrical about the \(y\)-axis and odd functions have rotation symmetry by \(\pi\) about the origin. They can serve as a reminder and check against over-generalisation: any general claim true of all odd functions, for example, must be true of \(y = x^{3}\) in particular.