Show that the Modulus Function f : R → R, given by f (x) =

1.	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.
Answer» Let x₁, x₂ ∈ R f (x.) = f (x₁) f(x₁) = f(x₂) ⇒ \|x₁\| = \|x₂\| ⇒ x₁ = x₂ \|-2\| = \|+2\| = 2 hence f is not one-one. Range of functions is only non negative real numbers Range of f = {0, ∞) ≠ R ∴ f is not onto.

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.

Answer»

Let x₁, x₂ ∈ R f (x.) = f (x₁)

f(x₁) = f(x₂) ⇒ |x₁| = |x₂|

⇒ x₁ = x₂

|-2| = |+2| = 2

hence f is not one-one.

Range of functions is only non negative real numbers

Range of f = {0, ∞) ≠ R

∴ f is not onto.