Determine if the following are functions. f: M_2(Z*) rightarrow Q by f([a 0 0 b]
ID: 3007829 • Letter: D
Question
Determine if the following are functions. f: M_2(Z*) rightarrow Q by f([a 0 0 b]) = a/b where M_2(Z*) = {[a c b d] a, b, c, d Z*} f: Z[i]* rightarrow Q by f(a + bi) = a/b where Z[i]* = {a + bi C*| a, b Z}. f: Q* rightarrow C by f(a/b) = a^2 + b^2/ab + a^2 - b^2/ab i. f: (Z^+ times Z^+) rightarrow Z^+ by f(a, b) = 2ab - a - b + 1 8. Determine if the following functions are 1-1 and/or onto. f: (R - {2}) rightarrow (R - {1}) by f(x) = x + 4/x - 2 f: C rightarrow [0, infinity) by f(a + bi) = (a^2 + b^2) Let f: Z rightarrow 2Z (events) by f(a) = a^3 + 3a + 2.Explanation / Answer
Rule is that if a relation is a function, then each and every value of its domain should be associated with a unique value of its codomain. Or in a function, no one value of domain should be associated with more than one value in codomain.
Now in first part the determinant of given matrix will return = a/b that will be a unique value for both a and b
So option a) is a function.
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Here given relation will be a function because for each complex number of a+bi, its output is a unique fraction of form a/b.
So clearly part b) relation is also a function.
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Here function in part c) is also a function because for each a and b, it will return a separate unique complex number of form a+bi.
So this is also a function.
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And last one is also a function, because for two different positive integers, the expresson
2ab- a-b+1 will always return a unique positive integer.
So answer of part d is also YES it is a function.
Thus all given are the functions.
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