Which Functions Have an Inverse? (Multiple Answers)

(1) \(f(x) = x^3\)

(2) \(f(x) = 2x + 1\)

(3) \(f(x) = e^x\) from \(\mathbb{R}\) onto \((0,\infty)\)

Explanation

\(x^3\): bijective on \(\mathbb{R}\) → has inverse \(\sqrt[3]{x}\). \(x^2\): not injective (\(f(2)=f(-2)\)) → no inverse. \(2x+1\): bijective → inverse \(\frac{x-1}{2}\). \(\sin x\): not injective nor surjective on \(\mathbb{R}\) → no inverse (it has one only when restricted to \([-\pi/2, \pi/2]\)). \(e^x\): bijective onto \((0,\infty)\) → inverse is \(\ln x\).