Give examples of two one-one functions f1 and f2 from R to R- such that f1-f2- R - R- defined by f1-f2x-f1x-f2x is not one-one-

We know that f1: R rarr; R, given by f1(x)=x, and f2(x)= x are one one. Proving f1 is one one: Let #160;f1x=f1y #8658;x=y So, f1 is one one. Proving f2 ...

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Byju's Answer

Standard XII

Mathematics

Differentiation under Integral Sign

Give examples...

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 f₁is one-one:
$Let f_{1} (x) = f_{1} (y) \Rightarrow x = y$
So, f₁is one-one.

Proving f₂is one-one:
$Let f_{2} (x) = f_{2} (y) \Rightarrow - x = - y \Rightarrow x = y$
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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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.

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.