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Question

Consider f : N → N, g : N → N and h : N → R defined as f(x) = 2x, g(y) = 3y + 4 and h(z) = sin z for all x, y, z ∈ N. Show that ho (gof) = (hog) of.

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Solution

Given, f : N → N, g : N → N and h : N → R
gof : N → N and hog : N → R
ho (gof) : N → R and (hog) of : N → R
So, both have the same domains.
gof x=g f x=g 2x=3 2x+4=6x+4 ...1hog x=hg x=h 3x+4=sin 3x+4 ... 2Now,h o gof x=h gof x=h6x+4 = sin 6x+4 [from 1]hog o f x=hog f x=hog 2x=sin 6x+4 [from 2]So, h o gof x=hog o f x, xNHence, h o gof=hog o f

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