Example 26 Consider a function f : 0, 2
π

→




R given by f (x) = sin x and
g : 0, 2
π

→




R given by g(x) = cos x Show that f and g are one-one, but f + g is not
one-one Solution Since for any two distinct elements x1 and x2 in 0, 2
π






, sin x1 ≠ sin x2 and
cos x1 ≠ cos x2, both f and g must be one-one But (f + g) (0) = sin 0 + cos 0 = 1 and
(f + g)
π2





 = sin
cos
1
2
2
π
π
+
=