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**NCERT mathematics class 12 chapter Relations and Functions Exercise 1.2 Problem 4 Solution**. Solutions for other problems are available at Exercise 1.2 SolutionsExercise 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_2 ∈ R 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 x ∈ R is mapped to a positive real number y ∈ R

⇒ There are few negative numbers y ∈ R in the co-domain, which are not images of any element in the domain R

∴ f is not onto.