Answer to Question #213804 in Discrete Mathematics for Aroosha ch

Question #213804

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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G;R     R. g(x)=x^3

                           


1
Expert's answer
2021-07-07T10:30:32-0400

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, x\\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..



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