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# Let f, g and h be functions from R to R. Show that (f + g) o h = foh + goh (f . g) o h = (foh) . (goh)

Q.2 Let $f$, $g$ and $h$ be functions from $R$ to $R$. Show that
$\\(f + g) o h = foh + goh\\ (f \cdot g) o h = (foh) \cdot (goh)$

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To prove : $\\(f + g) o h = foh + goh$

$((f + g) o h)(x)$

$=(f + g) ( h(x) )$

$=f ( h(x) ) +g(h(x))$

$=(f o h)(x) +(goh)(x)$

$=\left \{ (f o h) +(goh) \right \}(x)$                      $x\forall R$

Hence, $\\(f + g) o h = foh + goh$

To prove:$(f \cdot g) o h = (foh) \cdot (goh)$

$((f . g) o h)(x)$

$=(f . g) ( h(x) )$

$=f ( h(x) ) . g(h(x))$

$=(f o h)(x) . (goh)(x)$

$=\left \{ (f o h) .(goh) \right \}(x)$                      $x\forall R$

$\therefore$   $(f \cdot g) o h = (foh) \cdot (goh)$

Hence, $(f \cdot g) o h = (foh) \cdot (goh)$

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