# Definition:Vector-Valued Function

## Definition

Let $f_1, f_2, \ldots, f_n$ be real functions of $t$.

Let $\mathbb T \subseteq \R, \mathbb Y \subseteq \R^n$ (where usually $n \ge 2$).

Let $\mathbf r$ be a mapping from $\mathbb T \to \mathbb Y$ that maps each $t \in \mathbb T$ to a vector $\left \langle{f_1 \left({t}\right), f_2 \left({t}\right), \ldots, f_n \left({t}\right)}\right \rangle \in \mathbb Y$.

Then $\mathbf r$ is said to be a vector-valued function (of the parameter $t$).

If $\mathbb T$ is not explicitly defined, it is taken to be the intersection of all the domains of $f_1 ,f_2, \cdots, f_n$.

### Component Function

Each $f_1, f_2, \ldots, f_n$ is a component function of $\mathbf r$.