Test Index

AP DSC School Assistant Math Model Test 1

Section: Subject Specific

Question : 42 of 160

Marks: +1, -0

R

→

R

defined by f(x) = sinx is:

surjective but not injective
neither injective nor surjective
injective but not surjective
not a relation

Validate

Solution:

Concept:

A function is injective (one‑one) if different inputs produce different outputs. It is surjective (onto) if every element of the codomain is the image of at least one input.

Explanation:

Consider

f: \mathbb{R} \to \mathbb{R}

given by

f(x) = \sin x

Injectivity: $\sin x$ is periodic with period $2\pi$ , so many $x$ give the same output. For example, $\sin(\pi/2) = 1$ and $\sin(5\pi/2) = 1$ . Hence $f$ is not injective.

Surjectivity: The range of $\sin x$ is $[-1,1]$ , which is a proper subset of $\mathbb{R}$ . Numbers like $2$ or $100$ have no pre‑image. Thus $f$ is not surjective.

Relation: Every function is a special kind of relation, so $f$ is indeed a relation.

Therefore the function is neither injective nor surjective.

Answer:

Option B: neither injective nor surjective.

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