How to compose vector-valued functions

Question

Let $f(u,v) = (uv, u+v)$ and $g(x,y) = (e^{xy}, x-y)$. Calculate $f \circ g$. I don't understand how to compose these functions together. The question does not make any sense to me at all whatsoever, so I don't even know how to approach the problem.

Answer 1

If $f,g:\mathbb{R}^2\rightarrow \mathbb{R}^2$, then, exists $f\circ g:\mathbb{R}^2\rightarrow \mathbb{R}^2$, defined for $$(f\circ g)(x,y)=f(g(x,y))=f(e^{xy},x-y)=(e^{xy} (x-y), e^{xy}+x-y). $$

Answer 2

Function composition is a very general thing not at all limited to real functions. For example, any time I have $3$ sets $A,B$ and $C$ and two functions, $g:A\to B$ and $f:B\to C$ I can define a compositum $f\circ g$ like this:

$$f\circ g:A\to C\\ \forall a\in A: (f\circ g)(a) := f(g(a)) $$

This is a well defined function, because for every $a\in A$, $b=g(a)$ is a well defined element in $B$. For such an element $b$, the element $c=f(c)$ is also defined, so $f(g(a))$ is defined.

In your case, all three sets are the same, that is $A=B=C=\mathbb R^2$. You have functons $f$ and $g$ and can therefore calculate the function $f\circ g$ as per definition. Try it and post results, to see where the problem is. Then we can help you further.

Answer 3

What you do is just to replace the parameters for the outer function ($f$ in this case) with the parameters for the inner function ($g$ in this case). So it would be:

$f \circ g = (e^{xy}(x-y),e^{xy}+x-y)$