Now, gof(x) = g (f(x)) = g(4x + 3) = (4
3
3)
4
x
x
+
−
=
 and
fog(y) = f (g (y)) = f
(
3)
4 (
3)
3
4
4
y
y
−
−

 =
+




 = y – 3 + 3 = y This shows that gof = IN
and fog = IY, which implies that f is invertible and g is the inverse of f Miscellaneous Examples
Example 18 If R1
 and R2 are equivalence relations in a set A, show that R1 ∩ R2 is
also an equivalence relation Solution Since R1
 and R2 are equivalence relations, (a, a) ∈ R1, and (a, a) ∈ R2 ∀ a ∈ A