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 commutative.
Answer» False In order to prove composition of functions is commutative we need to show (fog)(x) = (gof)(x) Let us suppose, f(x) = 2x, g(x) = 1 + x² Now, (fog)(x) = f(g(x)) = f(1+x²) = 2(1+x²) = 2+2x² (i) (gof)(x) = g(f(x)) = g(2x) = 1+(2x)² = 1+4x² (ii) From (i) and (ii), we observe that (fog)(x) ≠ (gof)(x) Thus, composition of functions is not commutative.

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

Answer»

False

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

(fog)(x) = (gof)(x)

Let us suppose, f(x) = 2x, g(x) = 1 + x²

Now,

(fog)(x) = f(g(x)) = f(1+x²)

= 2(1+x²) = 2+2x² (i)

(gof)(x) = g(f(x)) = g(2x)

= 1+(2x)² = 1+4x² (ii)

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

(fog)(x) ≠ (gof)(x)

Thus, composition of functions is not commutative.