State True or False for the statement:The composition of functions is

1.	State True or False for the statement:The composition of functions is associative.
Answer» 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.

State True or False for the statement:The composition of functions is associative.

Answer»

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.