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Question
Show that if f1 and f2 are one-one maps from R to R, then the product f1 × f2 : R → R defined by (f1 × f2) (x) = f1 (x) f2 (x) need not be one - one.
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 x and y be two elements in the domain R, such that
f1(x) = f1(y)
⇒ x = y
So, f1 is one-one.
Proving f2 is one-one:
Let x and y be two elements in the domain R, such that
f2(x) = f2(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 = x2
Let x and y be two elements in the domain R, such that
( f1 × f2) (x) = (f1 × f2) (y)
⇒ x2 = y2
⇒ x = ± y
So, (f1 × f2) is not one-one.
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