Tests — Inverse circular functions
Arc sine, arc cosine and arc tangent (chapter VII).
Teste 158MACK-74 Let $f$, $g$ and $h$ be functions of $A$ in $A$, where $A = [-1, 1]$, defined as follows: $f(x) = \operatorname{sen} x$, $g(x) = \operatorname{sen}\pi x$, $h(x) = \frac{\pi}{2}x$. We can state that:
a) they are all invertibleb) they are all superjectivec) only one is injectabled) only one is surjectivee) only one is injective and surjective $f$ (sine in $[-1, 1]$) is injective, with image $[-\operatorname{sen} 1, \operatorname{sen} 1]$; $g(x) = \operatorname{sen}\pi x$ has image $[-1, 1]$ (surjective), but is not injective ($g(0) = g(1) = 0$); $h$ is injective and its image $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ doesn’t even fit into $A$. Only $g$ is surjective. The printed template indicates (e), but none of the three is both an injector and a surjector of $A$ in $A$; the true statement is (d). Teste 159MACK-73 The domain of the function defined by $y = \operatorname{arc\,sen}\sqrt{2x - 3}$ is:
a) $\left\{x \in \mathbb{R} \;\middle|\; x \ge \frac{3}{2}\right\}$b) $\left\{x \in \mathbb{R} \;\middle|\; \frac{3}{2} \le x \le 2\right\}$c) $\{x \in \mathbb{R} \mid 0 \le x \le 2\}$d) $\left\{x \in \mathbb{R} \;\middle|\; -2 \le x \le 0 \text{ ou } \frac{5}{2} \le x \le 4\right\}$e) none of the previous answers The root requires $2x - 3 \ge 0$ and the arcsine requires $\sqrt{2x - 3} \le 1$: $0 \le 2x - 3 \le 1$, that is, $\frac{3}{2} \le x \le 2$.
