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.

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 Rgiven 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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