Exercise · ZW-INJECTIVE-FUNCTION-002 · ★★☆ Intermediate
Which of these functions from \(\mathbb{R}\) to \(\mathbb{R}\) is injective?
Choose one option.
Show answer
- C. \(f(x)=2x+1\)
Explanation
With \(f(x)=2x+1\), \(2a+1=2b+1\) gives \(a=b\). The other three are not injective: \(x^2\) gives the same value at \(2\) and \(-2\), \(\lvert x\rvert\) at \(1\) and \(-1\), and \(\sin\) at \(0\) and \(\pi\), where \(\sin 0=\sin\pi=0\).
Review the concept: Injective function
Function \(f\colon A\to B\) in which distinct elements of \(A\) have distinct images in \(B\).
\[\forall a_1,a_2\in A\,\bigl(f(a_1)=f(a_2)\Rightarrow a_1=a_2\bigr)\]
More practice problems: injective function
- ZW-INJECTIVE-FUNCTION-001 What does it mean for fcolon A→ B to be injective?
- ZW-INJECTIVE-FUNCTION-003 What is the logical form of the definition of an injective function?
- ZW-INJECTIVE-FUNCTION-004 If fcolon A→ B is not injective, what holds?
- ZW-INJECTIVE-FUNCTION-005 Can there be an injective function fcolon\1,2,3\→\a,b\?
- ZW-INJECTIVE-FUNCTION-006 If f is injective and a_1ne a_2, what happens to f(a_1) and f(a_2)?
- ZW-INJECTIVE-FUNCTION-007 Let fcolonℤ→ℤ, f(n)=2n. Is f injective?
- ZW-INJECTIVE-FUNCTION-008 If fcolon A→ B and gcolon B→ C are injective, is g∘ f?
- ZW-INJECTIVE-FUNCTION-009 Is every injective function also surjective?