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Algebra Level 3

If the function f : R → R f: \mathbb R \to \mathbb R defined by f ( x ) = ∣ x − 2 ∣ f(x) = | x - 2 | , then which of a property of the function f f ?

Neither injective, Nor surjective Injective only Bijective Surjective only

Akash Patalwanshi by akash patalwanshi , 28, India

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2 solutions

f ( x ) = ∣ x − 2 ∣ f(x) = |x - 2| is not an injective function because f ( 3 ) = f ( 1 ) ∨ f ( 0 ) = f ( 4 ) f(3) = f(1) \space \vee \space f(0) = f(4) and it's not a superjective function due to there doesn't exist x ∈ R x \in \mathbb{R} such that f ( x ) = − 1 ∀ x ∈ R f(x) = - 1 \space \forall x \in \mathbb{R}

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Akash Patalwanshi
May 20, 2016

This absolute value function f \large f has y-values that are paired with more than one x-value, such as ( 4 , 2 ) (4, 2) and ( 0 , 2 ) (0, 2) Hence f \large f is not one-to-one i.e not injective.

In addition, values less than 0 0 on the y-axis are never used, making the function NOT onto. i. e. Not surjective.

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