Question

$f=\left\{\right(1,2),(3,5),(4,1\left)\right\}$ and $g=\left\{\right(1,3),(2,3),(5,1\left)\right\}$. Write down Let $f,g$ and $h$ be functions from $R$ to $R$. Show that
$(f+g)oh=foh+goh$
$(f.g)oh=\left(foh\right).\left(goh\right)$

Answer 1

Let $f,g$ and $h$ be a function from $R$ to $R$

(i) $(f+g)oh=foh+goh$

Let us consider $\left(\right(f+g\left)oh\right)\left(x\right)=(f+g)\left(h\right(x\left)\right)$

$=f\left(ch\right(x\left)\right)+g\left(g\right(x\left)\right)$

$=\left(foh\right)\left(x\right)+\left(goh\right)\left(x\right)$

$\left[\right(foh)+(goh\left)\right]\left(x\right)$

$\therefore \left(\right(f+g\left)oh\right)\left(x\right)=\left[\right(foh)+(goh\left)\right]\left(x\right)$

$\therefore (f+g)\left(oh\right)=\left(foh\right)+\left(goh\right)$

(ii) $(f\cdot g)oh=\left(foh\right)\left(goh\right)$

Let $x\in R$

Consider $\left(\right(f\cdot g\left)oh\right)\left(x\right)=(f\cdot g)\left(h\right(x\left)\right)$

$=f\left(h\right(x\left)\right)\cdot g\left(h\right(x\left)\right)$

$=\left(foh\right)\left(x\right)\left(goh\right)\left(x\right)$

$\left(\right(f\cdot g\left)oh\right)\left(x\right)=\left[\right(fog)\cdot (goh\left)\right]\left(x\right)$

$\therefore (f\cdot g)oh=\left(fog\right)\cdot \left(goh\right)$

$\therefore f+gog=\left(foh\right)+goh$

and $(f\cdot g)oh=\left(fog\right)\cdot goh$

Hence Proved