Exercise · ZW-BIJECTIVE-FUNCTION-006 · ★★☆ Intermediate
Is \(f\colon\mathbb{R}\to\mathbb{R}\), \(f(x)=x^3-x\) bijective?
Choose one option.
Show answer
- A. No, because \(f(0)=f(1)=0\) and \(0\neq1\)
Explanation
For \(f(x)=x^3-x\) we have \(f(0)=f(1)=0\) with \(0\neq1\), so it is not injective and therefore not bijective. Being surjective, as every real polynomial of odd degree is, is not enough, and being a polynomial guarantees nothing either.
Review the concept: Bijective function
Function \(f\colon A\to B\) that is both injective and surjective, so each element of \(B\) corresponds to exactly one element of \(A\).
\[\forall b\in B\;\exists!\,a\in A\,(f(a)=b)\]
More practice problems: bijective function
- ZW-BIJECTIVE-FUNCTION-001 Which two properties must fcolon A→ B satisfy to be bijective?
- ZW-BIJECTIVE-FUNCTION-002 If fcolon A→ B is bijective, how many elements of A have image equal to a given b∈ B?
- ZW-BIJECTIVE-FUNCTION-003 What does it mean for fcolon A→ B to be injective?
- ZW-BIJECTIVE-FUNCTION-004 Which is the definition of surjectivity for fcolon A→ B?
- ZW-BIJECTIVE-FUNCTION-005 Let fcolon\1,2,3\→\a,b,c\ be injective. What can be concluded?
- ZW-BIJECTIVE-FUNCTION-007 Which of these functions from ℤ to ℤ is bijective?
- ZW-BIJECTIVE-FUNCTION-008 If fcolon A→ B is bijective and |A|=5, what is |B|?
- ZW-BIJECTIVE-FUNCTION-009 Let fcolonℝ→ℝ, f(x)=x^3+x. Which statement is correct?