From Relations to Functions — Definition, Domain, Codomain and Range

In an earlier chapter a relation $R$ from a set $A$ to a set $B$ was defined as any subset of the Cartesian product $A \times B$ — that is, any chosen collection of ordered pairs $(a,b)$ with $a \in A$ and $b \in B$. A function is a relation that obeys one extra, very strict rule, and it is this rule that makes a function so useful for describing dependable, predictable relationships.

A relation $f$ from a set $A$ to a set $B$ is called a function (or a mapping) if every element of $A$ is related to exactly one element of $B$ — never left unpaired, and never paired with two different elements. We write $f : A \to B$, and if $a \in A$ is associated with $b \in B$ under $f$ we write $b = f(a)$ and call $b$ the image of $a$, and $a$ a pre-image of $b$.

Two conditions together define a function, and both must hold:

• Existence — every element of $A$ must have an image (no element of $A$ is left out).
• Uniqueness — no element of $A$ may have two or more different images.

Three sets accompany every function $f : A \to B$:

• The domain is the set $A$ itself — every element of $A$ must have an image, with no exceptions.
• The codomain is the set $B$ — the set from which images are drawn, whether or not every element of $B$ is actually used.
• The range (or image set) is the subset of $B$ made up only of those elements that are actually an image of some element of $A$, i.e. $\text{range}(f) = \{\, f(a) : a \in A \,\}$. The range is always a subset of the codomain, and the two are equal only in a special case (the "onto" functions of Section 2).

When a function is described only by a rule such as $y = f(x)$ with no domain stated explicitly, we take the domain to be the largest set of real numbers for which the rule gives a genuine real value — this is the natural domain. For example, $f(x) = \dfrac{1}{x-3}$ has natural domain $\mathbb{R} - \{3\}$, because the rule fails only where the denominator is zero.

A quick graphical way to test whether a curve drawn in the $xy$-plane represents $y$ as a function of $x$ is the vertical line test: the curve is a function of $x$ if and only if every vertical line meets it in at most one point. A circle fails this test — a vertical line through it meets it twice — and so a circle is not the graph of a function of $x$.

The whole of this chapter in the Maharashtra Std XI Mathematics and Statistics (Commerce) syllabus builds on this single definition — every later type of function, and every graph, is just a different way a rule can assign to each input exactly one output.