Q. 195.0( 1 Vote )
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.
Answer :
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 are two functions given by
f1(x) = x
f2(x) = x
From above function it is clear that both are One – One functions
Now, f1×f2 : R → R
given by
⇒ (f1×f2 )(x) = f1(x)×f2(x) = x2
⇒ (f1×f2 )(x) = x2
Also,
f(1) = 1 = f( – 1)
Therefore,
f is not One – One
⇒ f1×f2 : R → R is not One – One function.
Hence Proved
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