Question

Verify the associative law for the composite function of following three functions :

f : N → Z₀, f(x) = 2x

g : Z₀ → Q, g(x) = 1/x

h : Q → R, h(x) = e^x

Answer 1

Given,

f : N→ Z₀

g : Z₀ → Q

h : Q → R

Then, ho(gof) : N → R

and (hog)of : N → R

Hence, domain and co-domain of ho(gof) and go(hof) are same because both functions are defined from N to R.

Hence, we have to prove

[ho(gof)](x) = [(hog)of)(x), ∀ X ∈ N

Now, [ho(gof)](x)= h[(gof)(x)]

= h[g{f(x)]

= h[g(2x)]

From eqs. (i) and (ii),

(hog)of = ho(gof)

Hence, associativity of f, g, h is proved. Proved.