Show that the signum function f : R → R given by f (x) = {1, if x > 0; 0, if x = 0; 0, if x = 0} is neither one-one nor onto.

Solution:

Signum function is defined as a mathematical function that gives the sign of a real number.

According to the given problem:

f : R → R given by f (x) = {1, if x > 0; 0, if x = 0; 0, if x = 0}

f (1) = f (2) = 1,

Here 1 and 2 is greater than 0.

but 1 ≠ 2

⇒ f is not one-one.

f (x) takes only 3 values(1, 0, - 1) for the element - 2 in co-domain

R, there does not exist any x in domain R such that f (x) = - 2.

⇒ f is not onto.

The signum function is neither one-one nor onto

NCERT Solutions for Class 12 Maths - Chapter 1 Exercise 1.2 Question 5

Show that the signum function f : R → R given by f (x) = {1, if x > 0; 0, if x = 0; 0, if x = 0} is neither one-one nor onto

Summary: 

The signum function f : R → R given by f (x) = {1, if x > 0; 0, if x = 0; 0, if x = 0} is neither one-one nor onto. The Signum function is defined as a mathematical function that gives the sign of a real number