Functions-19
From Homeworkwiki
Show that the function f: Z → Z defined by f(x) = x2 + x for all x ∈ Z is a many-one function.
Solution: Let x, y ∈ Z. Then,
f(x) = f(y) => x2 + x = y2 + y
=>(x2 - y2) + (x - y) = 0
=>(x - y)(x + y + 1) = 0
=>(x - y)(x + y + 1) = 0
=>x = y or y = -x -1
Since, f(x) = f(y) does not provide the unique solution x = y but it also provides y = -x -1, this means that x ≠ y but f(x) = f(y)
when y = -x – 1. For example, if we put x = 1 in y = -x – 1, we obtain y = -2. This shows that 1 and -2 have the same image under f.
Hence, f is a many-one function.
