MHB Function Composition: Restrictions for Domains

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To define the composite functions \(G \circ F\), \(H \circ G\), \(H \circ (G \circ F)\), and \((H \circ G) \circ F\), specific restrictions on the domains of the functions \(F\), \(G\), and \(H\) are necessary. The image of \(F\) must be a subset of the domain of \(G\), and the image of \(G\) must be a subset of the domain of \(H\). This ensures that each function can accept the output of the preceding function in the composition. For example, if \(F(x) = x + 5\) and \(G(x)\) is defined for all real numbers, then the conditions are satisfied. Properly defining these composite functions requires careful consideration of the relationships between their domains and codomains.
Dustinsfl
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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.
 
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I'd say that in each case you'd need $\textrm{Im }F \subset D_G, \text{ I am }G \subset D_H, \text{ I am }G \circ F \subset D_H$ 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).
 
dwsmith said:
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.
 
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