Question

Let f.g,h: R + R be functions. Prove that the followings is true or not. If not show an example. a) (g+h) • f = (gºf) + (hof) b) f(g+h) = (fºg) + (f • h)

Answer 1

b)

$\text{Let }f=g=h=\sin x$

$f\circ (g+h)=(\sin x)\circ(\sin x+\sin x)=\sin(2\sin x)$

$f\circ g+f\circ h=(\sin x)\circ(\sin x)+(\sin x)\circ(\sin x)=2\sin(\sin x)$

Hence,

$f\circ(g+h)\neq f\circ g+f\circ h$

a)

Let F = 9+h

$\text{Then we have }(g+h)\circ f=F\circ f=F(f)=g(f)+h(f)=g\circ f+h\circ f$

Therefore,

$(g+h)\circ f=(g\circ f)+(h\circ f)$