[SOLVED]function composition

dwsmith

Well-known member
Given three functions $F,G,H$ what restrictions must be placed on their domains so that the following four composite functions can be defined?
$$G\circ F,\quad H\circ G,\quad H\circ (G\circ F),\quad (H\circ G)\circ F.$$

I need a hint or something.

Fantini

MHB Math Helper
I'd say that in each case you'd need $\textrm{Im }F \subset D_G, \text{ Im }G \subset D_H, \text{ Im }G \circ F \subset D_H$ and $\text{Im }F \subset D_{H \circ G}$.

Taking as example your other thread, we need $\text{Im }F \subset D_G$, where $F(x) = x+5$ and

$$G(x) = \begin{cases} \frac{|x|}{x}, & \text{if } x \neq 0 \\ 1, & \text{if } x= 0. \end{cases}$$

For $\text{Im }F$ to be contained in the domain of $G$, we need to investigate the cases where $F(x) \neq 0$ and $F(x)=0$. In particular, here we have $\text{Im }F = \mathbb{R}$ and $D_G = \mathbb{R}$ since it is defined everywhere (although it isn't continuous).

Sudharaka

Well-known member
MHB Math Helper
Given three functions $F,G,H$ what restrictions must be placed on their domains so that the following four composite functions can be defined?
$$G\circ F,\quad H\circ G,\quad H\circ (G\circ F),\quad (H\circ G)\circ F.$$

I need a hint or something.
Hi dwsmith, If $$G\circ F$$ is to be defined properly the co-domain of $$F$$ should be a subset of the domain of $$G$$. Therefore the only restrictions in defining the above mentioned compositions are,

$\mbox{codom }(F)\subseteq\mbox{dom }(G)\mbox{ and }\mbox{codom }(G)\subseteq\mbox{dom }(H)$

Kind Regards,
Sudharaka.