Consider the following functions The function f N rightarrow

Consider the following functions: The function f: N rightarrow N defined by f(x) = |x| The function g: N rightarrow R defined by g(x) = |x| The function h: N rightarrow N defined by h(x) = |x| The function k: Z rightarrow N defined by k(x) = |x| Is f = g? Explain your answer. Is f = h? Explain your answer. Is f = k? Explain your answer. Find f(1) and f({1}). Find g^-1({1, 2, 3}, g^-1 ({-1, 2, 3}, and g^-1({-1}. Find h({1, 2, 3}) and h^-1({ 1, 2, 3}. Find k^-1 ({1, 2, 3}.

Solution

1) No. They are different as they have different ranges (co-domains)

2) No. They differ at -1. f(-1)=|-1|=1, whereas h(-1)=-1

3) No. They have different domains.

4) f(1)=|1|=1. f({1}) is not defined as {1} (the set containing 1) is not a natural number so it doesn\'t lie in the domain of f.