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Question
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.
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Solution
We know that f1: R → R, given by f1(x)=x, and f2(x)=-x are one-one. Proving f1 is one-one:
Let f1 (x) = f1 (y)
⇒ x = y
So, f1 is one-one.
Proving f2 is one-one:
Let f2 (x)=f2 (y)
⇒ −x = −y
⇒ x = y
So, f2 is one-one.
Proving (f1 + f2) is not one-one:
Given:
(f1 + f2) (x) = f1 (x) + f2 (x)= x + (-x) =0
So, for every real number x, (f1 + f2) (x)=0
So, the image of ever number in the domain is same as 0.
Thus, (f1 + f2) is not one-one.
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