Exercise · ZW-BIJECTIVE-FUNCTION-009 · ★★★ Advanced
Let \(f\colon\mathbb{R}\to\mathbb{R}\), \(f(x)=x^3+x\). Which statement is correct?
Choose one option.
A It is bijective, because \(f(x)=f(-x)\) for all \(x\) B It is bijective: \(f'(x)=3x^2+1>0\) makes it strictly increasing, and since \(f(x)\to\pm\infty\), continuity gives every value C It is not bijective, because \(f(0)=0\) makes it non-injective D It is not bijective, because it is not surjective Show answer B. It is bijective: \(f'(x)=3x^2+1>0\) makes it strictly increasing, and since \(f(x)\to\pm\infty\), continuity gives every value← Back to the question
Explanation A positive derivative makes \(f\) strictly increasing, so it is injective; since \(f(x)\to\pm\infty\), the intermediate value theorem gives surjectivity. The zero of \(f\) does not repeat, because \(x^3+x=0\) has only the solution \(x=0\).
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