Domain, Co-domain and Range

Domain, co-domain and range — precise definitions

For a function $f : A \to B$:

• Domain: the set $A$ itself — every value of $x$ for which $f(x)$ is defined.
• Co-domain: the set $B$ — the set the images are chosen from. It may contain elements that are never actually used as an image.
• Range: the set of images that are actually produced, i.e. $\{f(x) : x \in A\}$. The range is always a subset of the co-domain, but the two need not be equal.

Finding the domain of an algebraic function

When a function is defined by a formula without an explicit set $A$ stated, its domain is taken to be the largest subset of $\mathbb{R}$ for which the formula gives a real, defined value. Two situations to check for every time:

Finding the range

The range is usually found by asking: for which values of $y$ does the equation $y = f(x)$ have at least one real solution $x$ in the domain? For simple functions this can often be reasoned out directly (e.g. $x^2 \geq 0$ always, so $y \geq 0$); for others, solving the equation for $x$ in terms of $y$ and checking which $y$ keep $x$ real and in the domain is the reliable method. …