Give examples of two one – one functions f1 and f2 from R to R such that f1 + f2 : R → R, defined by (f1 + f2)(x) = f1(x) + f2(x) is not one – one.
TIP: – One – One Function: – A function is said to be a one – one functions or an injection if different elements of A have different images in B.
So, is One – One function
⇔ a≠b
⇒ f(a)≠f(b) for all
⇔ f(a) = f(b)
a = b for all
Let, f1: R → R and f2: R → R be two functions given by (Examples)
f1(x) = x
f1(x) = – x
From above function it is clear that both are One – One functions
Now,
⇒ (f1 + f2)(x) = f1(x) + f2(x)
⇒ (f1 + f2)(x) = x – x
⇒ (f1 + f2)(x) = 0
Therefore,
f1 + f2 : R → R is a function given by
(f1 + f2)(x) = 0
Since f1 + f2 is a constant function,
Hence it is not an One – One function.