We have explored functions of the form $$y=f(x)$$ closely throughout this text. We have explored their limits, their derivatives and their antiderivatives; we have learned to identify key features of their graphs, such as relative maxima and minima, inflection points and asymptotes; we have found equations of their tangent lines, the areas between portions of their graphs and the $$x$$-axis, and the volumes of solids generated by revolving portions of their graphs about a horizontal or vertical axis.
Despite all this, the graphs created by functions of the form $$y=f(x)$$ are limited. Since each $$x$$-value can correspond to only 1 $$y$$-value, common shapes like circles cannot be fully described by a function in this form. Fittingly, the “vertical line test” excludes vertical lines from being functions of $$x\text{,}$$ even though these lines are important in mathematics.