# Functions-19

### From Homeworkwiki

**Show that the function f: Z → Z defined by f(x) = x ^{2} + x for all x ∈ Z is a many-one function.**

**Solution:** Let x, y ∈ Z. Then,

f(x) = f(y) => x^{2} + x = y^{2} + y

=>(x^{2} - y^{2}) + (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.