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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.

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

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    $\begingroup$ 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 $m$. $\endgroup$ – André Nicolas Feb 2 '13 at 7:24
  • $\begingroup$ So the components are points? So what we have is actually $$\sin(x) = (x, \sin(x))$$? $\endgroup$ – Hawk Feb 2 '13 at 7:29
  • $\begingroup$ 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 $(x,f(x))$. $\endgroup$ – André Nicolas Feb 2 '13 at 7:45
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    $\begingroup$ 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 $C=\{(x,\;y)\in\mathbb{R}^2\colon\;\;x^2+y^2=1 \}$. $\endgroup$ – M. Strochyk Feb 2 '13 at 8:11

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 $f(x)=(f_1(x),f_2(x))$.


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