The function f: R rightarrow R defined by f(x) = x3 is onto because for any real

Question

The function f: R rightarrow R defined by f(x) = x3 is onto because for any real number r, we have that 3 r is a real number and f(3 r) = r . Consider the same function defined on the integers g: z rightarrow z by g(n) = n3. Explain why g is not onto z and give one integer that g cannot output. Let A = {x| x is a nation} B = {Asia, Africa, North America, South America, Antarctica, Europe, Australia} Let f:ArightarrowB be the function that outputs the continent to which a nation belongs. For example, f(lceland) = Europe and f(Greenland)= North America. Explain why f is not an Qnto function and give an example to prove it.

Explanation / Answer

4) since, g is defined on integers g(n)=n^3, if it is onto then for every integer in the Co-domain there should be a inverse map to the domain but it is not true,

Example. consider the integer 7 . cuberoot(7) is the inverse map . But Cuberoot(7) doesnt not exist in the domain.

Hence, it is not Onto.

5) If is has to be Onto then all the elements in B must be mapped . But only Europe and North America have been mapped. Hence, It is not onto.

Example, the element south america In B doesnt have a mapping to A.

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