Question

Composition of function is associative ____

a) Always true b) Never true c) Sometime true d) Not defined
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Answer 1

Answer:

always true

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Answer 2

Answer:

Option (a) is correct.

Step-by-step explanation:

Composition of a function is associative always true.

Proof:

Let us consider the functions as follows:

f : X → Y,  g : Y → Z and h : Z → W

Then, the compositions of the functions f, g, and h are:

g ∘ f : X → Z and h ∘ g : Y → W

To show: h ∘ (g ∘ f) = (h ∘ g) ∘ f

Consider the left-hand side as follows:

h ∘ (g ∘ f)(x) = h(g ∘ f(x)) (Applying x on the function f)

                   = h(g(f(x)))

                   = h ∘ g(f(x))

                   = (h ∘ g) ∘ f

Since the right-hand side is,

(h ∘ g) ∘ f

Thus, LHS = RHS

Therefore, the composition of the function is associative is always true.

Option (a) is correct.

Rest of the other options are incorrect.

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