Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

enter image description here

Can someone tell me what is meant by The action of f on input x is written out in component form is $f(x) = (f_1 (x), \dots, f_m(x) )$

What is the component of a function? So if $m = 2$, and say $f(x) = \sin(x)$, what are its components? I am not asking for the partials here.

$$\sin(x) = (\sin(x),\sin(y))$$

share|cite|improve this question
In the case $m=2$, the function maps points $x$ to points in the plane. The first component is the $x$-coordinate, the second component is the $y$-coordinate. Similar considerations apply to $m=3$ (we get a curve in space), and to bigger $m$. – André Nicolas Feb 2 '13 at 7:24
So the components are points? So what we have is actually $$\sin(x) = (x, \sin(x))$$? – Hawk Feb 2 '13 at 7:29
For $m=2$, we are talking about functions that map the reals to the plane. So $\sin x$ does not qualify, though its close relative that takes $x$ to $(\sin x,0)$ does. So does the function that takes $x$ to $(x,\sin x)$. But there are many other kinds, like spirals, that are not of shape $(x,f(x))$. – André Nicolas Feb 2 '13 at 7:45
Example: Let \begin{gather} U=[0,\; 2 \pi), \\ \begin{cases} f_1(t)=\cos{t}, \\ f_2(t)=\sin{t}. \end{cases} \end{gather} Then $f(t)=\left(f_1(t),\;f_2(t)\right)$ is a vector-function with components $f_1(t)$ and $f_2(t)$, which maps every point $t\in {U}$ to the point $(f_1(t),\; (f_2(t)) \in \mathbb{R}^2$ on the unit circle $C=\{(x,\;y)\in\mathbb{R}^2\colon\;\;x^2+y^2=1 \}$. – M. Strochyk Feb 2 '13 at 8:11
up vote 4 down vote accepted

Functions don't have components, vectors (or points) have components.

The function $f$ maps elements of $U$ to points in $\mathbb R^n$. "The action of $f$ on input $x$" just means what $f$ does to $x$, i.e. the output $f(x)$, which is a point in $\mathbb R^n$. Maybe it'll be clearer if we give this point a name, $y=f(x)$. Then $y$ can be "written out in component form" as $(y_1,y_2,\ldots,y_n)$. We can go further and treat $y_1, y_2, \ldots, y_n$ themselves as $n$ different functions of $x$, namely $f_1,f_2,\ldots,f_n$ which each map an element of $U$ to a number in $\mathbb R$.

For example, suppose $f:\mathbb R\to\mathbb R^2$ is the function that maps $x$ to the point $(\cos x,\sin x)$. The two components of this point are $\cos x$ and $\sin x$. We can define two functions $f_1(x)=\cos x$ and $f_2(x)=\sin x$, and then say that $f(x)=(f_1(x),f_2(x))$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.