Q. 103.8( 5 Votes )

Let f : N N : f(x) = 2x, g : N N : g(y) = 3y + 4 and h : N N : h(z) = sin z. Show that h o (g o f ) = (h o g) o f.

Answer :

To show: h o (g o f ) = (h o g) o f


Formula used: (i) f o g = f(g(x))


(ii) g o f = g(f(x))


Given: (i) f : N N : f(x) = 2x


(ii) g : N N : g(y) = 3y + 4


(iii) h : N N : h(z) = sin z


Solution: We have,


LHS = h o (g o f )


h o (g(f(x))


h(g(2x))


h(3(2x) + 4)


h(6x +4)


sin(6x + 4)


RHS = (h o g) o f


(h(g(x))) o f


(h(3x + 4)) o f


sin(3x+4) o f


Now let sin(3x+4) be a function u


RHS = u o f


u(f(x))


u(2x)


sin(3(2x) + 4)


sin(6x + 4) = LHS


Hence Proved.


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