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

Solution:

The modulus of a function, which is also called the absolute value of a function gives the magnitude and absolute value of a number irrespective of the number is positive or negative.

According to the given problem:

f : R → R given by f (x) = |x| = {x, if x ≥ 0 and - x, if x < 0}

f (- 1) = |- 1| = 1

and f (1) = |1| = 1

⇒ f (- 1) = f (1) but - 1 ≠ 1

⇒ f is not one-one.

Consider - 1 ∈ R

f (x) = |x| is non-negative.

There exist any element x in domain R such that

f (x) = |x| = - 1

⇒ f is not onto.

The modulus function is neither one-one nor onto

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

Summary:

Here we have shown that f is not one-one as well as not onto. Therefore, The modulus function is neither one-one nor onto