# Step By Step Calculus » 4.1 - Function Definition

Synopsis
• Given two sets, D_fD_f and E_fE_f, a functionff is a mapping from D_fD_f to E_fE_f that assigns to each element xx in D_fD_f a corresponding element y = f(x) y = f(x) (the image) in E_fE_f. An element y \in E_{f}y \in E_{f} may be the image of any number of elements in D_{f}D_{f}, but an element x \in D_{f}x \in D_{f} can only be mapped to a single image in E_{f}E_{f}. In other words, f(x)f(x) is a single element of E_{f}E_{f}.
• Given y\in E_fy\in E_f, the pre-image of yy is the set of points \{x\in D_f\;|\;f(x)=y\}\{x\in D_f\;|\;f(x)=y\}. This pre-image can have multiple points in it.
• The domain of a function ff is the set of values in D_fD_f for which ff is defined.
• The range of a function ff is the set E_fE_f of images f(x)f(x). In mathematical notation the range of ff is the set \{y\;|\;\exists x \{y\;|\;\exists x such that f(x)=y\} f(x)=y\}.
• The codomain of a function ff is any set Y_fY_f that contains the range. In notation, Y_f\supseteq E_fY_f\supseteq E_f, where the range E_f=\{y\;|\;\exists x E_f=\{y\;|\;\exists x such that f(x)=y\} f(x)=y\}.
• The graph of a function is the set of points \{(x,f(x))\;|\;x\in D_f\}\{(x,f(x))\;|\;x\in D_f\} in the Cartesian plane. A curve represents the graph of a function if it passes the vertical line test: No vertical line intersects the curve at more than one point.
• Functions f:D_f \subseteq \mathbb{R} \rightarrow E_f \subseteq \mathbb{R}f:D_f \subseteq \mathbb{R} \rightarrow E_f \subseteq \mathbb{R} are often specified without giving the domain and range. In such cases, the domain D_fD_f is taken to be the largest subset D_fD_f of \mathbb{R}\mathbb{R} such that f(x)f(x) makes sense for each x\in D_fx\in D_f, while the range is the subset E_fE_f of \mathbb{R}\mathbb{R} to which ff maps its domain.