Problem 4 Solution

This page contains the NCERT mathematics class 12 chapter Relations and Functions Exercise 1.2 Problem 4 Solution. Solutions for other problems are available at Exercise 1.2 Solutions
Exercise 1.2 Problem 4 Solution
4. Show that the Modulus Function f : R → R, given by {f(x) = |x|}, is neither one-one nor onto, where |x| is x, if x is positive or 0 and |x| is –x, if x is negative.
To check whether f is one-one:
The function f : R → R is defined as {f(x) = |x|}
Consider two elements x_1, x_2R in the domain R such that
{f(x_1) = f(x_2)}
{|x_1| = |x_2|}
x_1 = \pm x_2
⇒ Two different elements, one positive and one negative but equal in magnitude have the same image in the co-domain R.
f is not one-one.
To check whether f is onto:
The function f : R → R is defined as {f(x) = |x|}
As we know |x| is always positive
⇒ Every element xR is mapped to a positive real number yR
⇒ There are few negative numbers yR in the co-domain, which are not images of any element in the domain R
f is not onto.