Which of the following functions is onetoone Which is onto a

Which of the following functions is one-to-one? Which is onto?

a. f : N N f (m) = m + 2

b. g : Z Z g(m) = 2m2 7.

Solution

a)

The function f is one to one but not onto

Let the function is not one-one that implies

f(m1) = f(m2), where m1 is not equal to m2

m1 + 2 = m2 + 2

m1 = m2

hence this gives the contradiction since we assumed that m1 is not equal to m2, hence the function is one-one

The function is not onto since the number {1,2} belong to range but they don\'t have any image since all the natural numbers are not covered

b)

Let the function is not one-one that implies

g(m1) = g(m2), where m1 is not equal to m2

2m1^2 - 7 = 2m2^2 - 7

m1^2 = m2^2

Hence -1 and 1 will map to the same number, hence the function is not one-one

The function is not onto since the number 2m^2 will always be positive hence the minimum which function can take is -7, hence the function can\'t take any values like -8,-9 and so on, hence the complete Z range is not covered

Therefore, the function g is neither one-one nor onto