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.

Answer 1

We know that f₁: R → R, given by f₁(x)=x, and f₂(x)=-x are one-one.
Proving f₁ is one-one:
$$\begin{gathered} \mathrm{Let}{f}_1\left(x\right)=f_1\left(y\right) \\ \Rightarrow x=y \end{gathered}$$
So, f₁ is one-one.

Proving f₂ is one-one:
$$\begin{gathered} \mathrm{Let}{f}_2\left(x\right)=f_2\left(y\right) \\ \Rightarrow -x=-y \\ \Rightarrow x=y \end{gathered}$$
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.