Show that if f 1 and f 2 are one – one maps from R to R, then the product f 1 × f 2 : R → R defined by (f 1 × f 2 )(x) = f 1 (x)f 2 (x) need not be one – one.

Question

Philoid · Accepted 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 → 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

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.

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.