State True or False for the statements

The composition of functions is associative.

True

In order to prove composition of functions is associative we need to show

[fo(goh)](x) = [(fog)oh](x)

Let us suppose, f(x) = x, g(x) = 2x, h(x) = x + 2

Now,

[fo(goh)](x) = f(g(h(x))) = f(g(x+2)) [∵ h(x) = x + 2 ]

= f(2(x+2)) = f(2x+4)

= 2x+4 (i)

[(fog)oh](x) = (fog)oh(x) = (fog)(h(x))

= (fog)(x+2) = f(g(x+2))

= f(2(x+2)) = f(2x+4)

= 2x+4 (ii)

From (i) and (ii), we observe that

[fo(goh)](x) = [(fog)oh](x)

Thus, composition of functions is associative.