Composition of Functions from $\mathbb{R}$ to $\mathbb{R}^2$

Question

Is the composition of functions from $\mathbb{R}$ to $\mathbb{R}^2$ a well defined notion? I was asked whether or not the composition of such functions constitutes a binary operation, but I don't know of a standard way to compose such functions!

Answer 1

No, in general you can only compose functions when the range of the first is the domain of the second.

If $f:M\rightarrow N$ and $ g: N\rightarrow P$ then $g\circ f$ makes sense.

With $f1, f2 : \mathbb{R} \rightarrow \mathbb{R}^2 $

The composition doesn't make any sense unless you also have some sort of projection operator $\pi : \mathbb{R}^2 \rightarrow \mathbb{R}$ in between.

For example let

$f1(x) = (x^2,17-x)$

and

$f2(x) = (3x^3,\sqrt{x})$

and

$\pi(x,y) = x-y$

Then

$f1 \circ \pi \circ f2(x) = ((3x^3 - \sqrt{x})^2, 17 - (3x^3 - \sqrt{x}))$

This is well defined. But only after one specifies some in between function to take us back to $\mathbb{R}$.

Answer 2

Hint: If $f : \mathbb{R} \to \mathbb{R}^2$ and $g : \mathbb{R} \to \mathbb{R}^2$ are two functions, then given any $x \in \mathbb{R}$, $f(x) = (a,b)$ for some $a \in \mathbb{R}$ and $b \in \mathbb{R}$. Given that $g$ must take real numbers as its arguments, is it therefore possible to define $g(f(x))$?