Which functions in Exercise 12 are onto?

Answer:Step-1: In this problem we need to determine whether each of the given functions from Z to Z is onto (or) not. Given : a) f(n) = n - 1 b) c) d) Onto function:A function is said to be onto if for every y in Y, there is an x in X such that f(x) = y.Test for onto functions: Replace f(x) by y and find Out the x value in terms of y. Substitute it in f(x) and solve .If we get y as a result , that is there exist an x such that f(x) = y then the function f would be onto , otherwise not.a)In this problem we need to determine f(n) = n-1 is onto (or) not. Given : f(n) = n - 1 Let us consider , f(n) = y. So , y = n- 1 Therefore , f(y+1) = y +1 - 1 , since f(n) = n -1. = y. Therefore , the given function is onto.Step-2: b)In this problem we need to determine is onto (or) not. Given : Let us consider , f(n) = y. So , Therefore , , since . = y. And , since . = y. By the definition of onto function is not ontoTherefore , the given function is not onto.Step-3: c)In this problem we need to determine is onto (or) not. Given : Let us consider , f(n) = y. So , By the definition of onto function is not onto.Because is not an integer. Therefore , the given function is not onto.Step-4: b)In this problem we need to determine is onto (or) not. Given : is an onto function . Because for every integer y there exist an integer n such that f(n) = y. That is , f(n) = y if and only if . Therefore , the given function is onto function.