Answer to Question #213920 in Discrete Mathematics for Danish

Question

Find each of the function below, indicate whether the function in onto, on-to-one neither or both. If the function is not onto or nor one-to-one, give an example showing why

G;R      R. g(x)=x^3

Answer 1

Consider the function $g:\R\to\R,\ g(x)=x^3.$ Let $g(x_1)=g(x_2).$ Then $x_1^3=x_2^3,$ and hence $x_1=x_2.$ We conclude that this function is one-to-one. For any $y\in\R$ the equation $x^3=y$ has a unique solution $x=\sqrt[3]{y},$ and hence for any $y\in\R$ there exists $x\in \R$ such that $f(x)=y.$ It follows that the function $f$ is onto. Therefore, this function is a bijection.