Question

Show that the Signum Function f: R → R, given by

f(x) = { 1, if x > 0
0 , if x = 0
-1 if x < 0 }

is neither one-one nor onto.

Answer 1

Given that f:R→R , given by
$f(x)=\left\{\begin{array}{ll}1&x>0\\0&x=0\\-1&x<0\end{array}\right.$

We can observed that f(1) = f(2) = 1 but 1 ≠ 2.
Thus, Signum function f is not one – one.

as you see, for any value of x , f(x) gives only three values e.g., 1 , 0 , -1
means, range of Signum function is {1, 0, -1}
but here given, co - domain belongs to R
e.g., Co - domain ≠ range
so, it is not onto function or surjective mapping.

hence, Signum function is neither one - one nor onto function.