AP DSC School Assistant Math Model Test 3

Concept:A function is injective (one‑one) if different inputs give different outputs. It is surjective (onto) if every real number is an output. The sine function oscillates between –1 and 1 and repeats values periodically.Explanation:Let $f : \mathbb{R} \to \mathbb{R}$ with $f(x) = \sin x$.1. Check injectivity: The sine function is periodic with period $2\pi$. Many distinct $x$‑values yield the same sine. For example, $\sin(\pi/2) = 1$ and $\sin(5\pi/2) = 1$. Since two different inputs give the same output, $f$ is not injective.2. Check surjectivity: The range of $\sin x$ is $[-1, 1]$. Any real number $y$ outside this interval (e.g., $y = 2$) has no $x \in \mathbb{R}$ such that $\sin x = y$. Therefore, $f$ is not surjective.3. By definition, every function is a relation; the question’s option D is irrelevant because $f$ is indeed a relation.Hence the function is neither injective nor surjective.Answer:Option B: neither injective nor surjective