Injective, surjective and bijective functions

Summary

Injective functions

• \(f: A→B\) is a given function.
• \(f\) is said to injective or one-to-one if it never maps distinct elements to the same element (horizontal line test).
• Formally, \(f\) is injective if for all \(x_1,x_2∈A\) , \(f\left( x_1 \right)=f\left( x_2 \right)⟹x_1=x_2\)
• If \(A⊆\) \(R\) , \(B⊆R\) and \(f\) is strictly increasing or strictly decreasing over its entire domain then \(f\) is injective.
• Examples:
• \(f:R→R\) defined by \(f(x)=2x\) is injective as it is strictly increasing.
• \(f:R→R\) defined by \(f(x)=x^2\) is not injective, \(f\left( -1 \right)=f\left( 1 \right)=1\) .
• \(f:[0,∞)→R\) defined by \(f(x)=x^2\) is injective. \(f\) is strictly increasing when \(x≥0\) .

Surjective functions

• \(f\) is said to surjective or onto if \(f\) reaches every element in the codomain
• Formally, \(f\) is surjective if for all \(y∈B\) there exists an \(x∈A\) such that \(f(x)=y\) .
• If the codomain is selected to be the range of \(f\) then \(f\) is always surjective.
• Examples:
• \(f:R→R\) defined by \(f(x)=2x\) is surjective. For any \(y∈R\) let \(x=y/2 \) and \(f\left( x \right)=y\) .
• \(f:R→R\) defined by \(f(x)=x^2\) is not surjective. There is no \(x∈R\) such that \(x^2=-1\) .
• \(f:R→[0,∞)\) defined by \(f(x)=x^2\) is surjective. For any \(y∈[0,∞)\) let \(x=\sqrt{y}\) and \(f\left( x \right)=y\) .

Bijective functions

• If \(f\) is both injective and surjective we say that \(f\) is bijective or a one-to-one correspondence .
• Examples:
• \(f:R→R\) defined by \(f\left( x \right)=ax+b\) is bijective whenever \(a≠0\) .
• \(f:[0,∞)→[0,∞)\) defined by \(f(x)=x^2\) is bijective.
• \(f:[0,∞)→[0,∞)\) defined by \(f(x)=\sqrt{x}\) is bijective.
• \(f:R→(0,∞)\) defined by \(f(x)=e^x\) is bijective.
• \(f:(0,∞)→R\) defined by \(f(x)=log x\) is bijective.