Answer to Question #213805 in Discrete Mathematics for Aroosha ch

Question #213805

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

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H;Z     Z. h(x)=x^3


1
Expert's answer
2021-07-07T15:06:44-0400

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


y1=y2=>x13=x23y_1=y_2=>x_1^3=x_2^3

=>(x1x2)(x12+x1x2+x22)=0=>(x_1-x_2)(x_1^2+x_1x_2+x_2^2)=0

=>x1=x2,wherex1,x2R=>x_1=x_2, \text{where} \,x_1,x_2\in \R

The function g(x)=x3g(x)=x^3 is one-to-one.


yR x=y3\forall y\in \R \ \exist x=\sqrt[3]{y} ​such that y=x3.y=x^3.

The function g(x)=x3g(x)=x^3 is onto.


Therefore the function g(x)=x3g(x)=x^3 is bijection.



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