Determine whether the function [tex]$f(x)=\frac{1}{x}$[/tex], defined on [tex]$R \{0\}$[/tex], is injective, surjective, and/or bijective. Justify your answer.
Answer (2)
The function f ( x ) = x 1 defined on R ∖ { 0 } is injective because different inputs map to different outputs. It is surjective because it covers every value in its codomain R ∖ { 0 } . Therefore, it is bijective as it is both injective and surjective.
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Show that if f ( x 1 ) = f ( x 2 ) , then x 1 = x 2 . Since x 1 1 = x 2 1 implies x 1 = x 2 , the function is injective.
Show that for every y in the codomain R \ { 0 } , there exists an x in the domain R \ { 0 } such that f ( x ) = y . Since x = y 1 is in R \ { 0 } for every y in R \ { 0 } , the function is surjective.
Because the function is both injective and surjective, the function is bijective.
The function f ( x ) = x 1 is injective, surjective, and bijective. bijective
Explanation
Problem Analysis We are given the function $f(x) =
\frac{1}{x} d e f in e d o n R \backslash {0}$, and we want to determine if it is injective, surjective, and/or bijective.
Proof of Injectivity To prove injectivity, we need to show that if f ( x 1 ) = f ( x 2 ) , then x 1 = x 2 . Suppose f ( x 1 ) = f ( x 2 ) . Then we have
x 1 1 = x 2 1
Multiplying both sides by x 1 x 2 , we get x 2 = x 1 . Thus, x 1 = x 2 , which means f ( x ) is injective.
Proof of Surjectivity To prove surjectivity, we need to show that for every y in the codomain R \ { 0 } , there exists an x in the domain R \ { 0 } such that f ( x ) = y . Let y ∈ R \ { 0 } . We want to find an x such that f ( x ) = y , i.e.,
x 1 = y
Solving for x , we get x = y 1 . Since y = 0 , x = y 1 is a real number and x = 0 . Thus, x ∈ R \ { 0 } . Therefore, for every y ∈ R \ { 0 } , there exists an x = y 1 ∈ R \ { 0 } such that f ( x ) = y . This means f ( x ) is surjective.
Conclusion Since f ( x ) is both injective and surjective, it is bijective.
Examples
Understanding injective, surjective, and bijective functions is crucial in cryptography. For example, the function f ( x ) = x 1 can be used in simple encryption algorithms where each input x is uniquely mapped to an output y . The bijective property ensures that each encrypted message can be uniquely decrypted back to its original form, maintaining the integrity and security of the communication.
