Answer to Question #233433 in Discrete Mathematics for kannan

Question

whether each of these functions is a bijection from R to R. f (x) = x 3 -1 

Answer 1

Consider the function $f:\R\to\R,\ f(x)=x^3-1.$ Let us show that this function is an injection. Let $f(x)=f(y).$ Then $x^3-1=y^3-1,$ and hence $x^3=y^3.$ It follows that $x=y,$ and $f$ is an injective function. Further, let us show that this function is a surjection. Since for any $y\in\R$ the equation $f(x)=y$ is equivalent to $x^3-1=y,$ and hence has the solution $x=\sqrt[3]{y+1}\in\R,$ we conclude that $f$ is a surjection. Consequently, the function $f$ is a bijection.