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Let $f$, gand h befunctions from R to R.Showthat
$$
(f+g) o h=f o h+g o h,(f \cdot g) o h=(f o h) \cdot(g 0 h)
$$
$$
(f+g) o h=f o h+g o h,(f \cdot g) o h=(f o h) \cdot(g 0 h)
$$
Solution:
1987 Upvotes
Verified Answer
$\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}, \mathrm{g}: \mathrm{R} \rightarrow \mathrm{R}, \mathrm{h}: \mathrm{R} \rightarrow \mathrm{R}$
(i) $(f+g) o h(x)=(f+g)[h(x)]=f[h(x)]+g[h(x)]$ $=\{$ foh $\}(x)+\{g o h\}(x)$
$\Rightarrow(f+g) o h=$ foh $+$ goh
(ii) $(\mathrm{f} \cdot \mathrm{g})$ oh $(\mathrm{x})=(\mathrm{f} \cdot \mathrm{g})[\mathrm{h}(\mathrm{x})]=\mathrm{f}[\mathrm{h}(\mathrm{x})] \cdot \mathrm{g}[\mathrm{h}(\mathrm{x})]$
$=\{$ foh $\}(\mathrm{x}) \cdot\{$ goh $\}(\mathrm{x})$
$\Rightarrow(f \cdot g) o h=(f o h) \cdot(g o h)$
(i) $(f+g) o h(x)=(f+g)[h(x)]=f[h(x)]+g[h(x)]$ $=\{$ foh $\}(x)+\{g o h\}(x)$
$\Rightarrow(f+g) o h=$ foh $+$ goh
(ii) $(\mathrm{f} \cdot \mathrm{g})$ oh $(\mathrm{x})=(\mathrm{f} \cdot \mathrm{g})[\mathrm{h}(\mathrm{x})]=\mathrm{f}[\mathrm{h}(\mathrm{x})] \cdot \mathrm{g}[\mathrm{h}(\mathrm{x})]$
$=\{$ foh $\}(\mathrm{x}) \cdot\{$ goh $\}(\mathrm{x})$
$\Rightarrow(f \cdot g) o h=(f o h) \cdot(g o h)$
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