Answer to Question #213805 in Discrete Mathematics for Aroosha ch

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

                               .           

H;Z     Z. h(x)=x^3

Answer 1

Notice that this function passes BOTH a vertical line test and a horizontal line test.

$$y_1=y_2=>x_1^3=x_2^3$$

$$=>(x_1-x_2)(x_1^2+x_1x_2+x_2^2)=0$$

$$=>x_1=x_2, \text{where} \,x_1,x_2\in \R$$

The function $g(x)=x^3$ is one-to-one.

$\forall y\in \R \ \exist x=\sqrt[3]{y}$ ​such that $y=x^3.$

The function $g(x)=x^3$ is onto.

Therefore the function $g(x)=x^3$ is bijection.