Trigonometry Demystified, 2nd edition Stan Gibilisco Explanations for Quiz Answers in Chapter 4 1. The inverse trigonometric functions, such as the Arcsine, operate on plain real numbers (the independent variable or "input") to get angles (the dependent variable or "output"). The expression "Arcsin 90º" has no meaning, because 90º is an inappropriate input quantity. The correct answer is D. 2. In Fig. 4-17, the uppermost interval shown (gray curve) corresponds to the closed interval [π/2,3π/2] for the dependent variable y. This interval encompasses all possible values of the independent variable x, while not producing more than one value for y at any point. The correct choice is A. All three of the other choices (B, C, and D) describe intervals that encompass all possible values of x, but all three of those intervals produce two values of y at most points, disqualifying them as true function curves. 3. Let's look at each of the three choices, and individually analyze them. When we take the inverse of the functionf (x) = x as stated in choice A, we get the relation f  -1 (x) = x which constitutes a true function (it's the same thing as the original). When we take the inverse of the function g (y) = 1/y as stated in choice B, we get the relation g -1 (y) = 1/y which, again, constitutes a true function because it's the very same thing as the original. When we take the inverse of the function h (z) = -z as stated in choice C, we get the relation h -1 (z) = -z which, yet again, is the very same thing as the original function! We've determined that all three choices A, B, and C will work, so the answer to this question is D, "All of the above." If you're astute, you'll notice that we don't have to exclude 0 from the domain for the functions f and h, although we must do so for g. Nevertheless, taking 0 out of the domains of f and h does no harm. 4. We can input any real number to the sine function and get a single, meaningful output value. Therefore, the domain of the sine function extends over the entire set of real numbers. The correct choice is A. 5. This question contains a quirk, so beware! Each equation expresses the value of the independent variable x in terms of the value of the dependent variable y. That's an "inside-out" portrayal, and isn't the usual way in which you'd express a function. Actually, choice A is the equivalent of the relationy = arccos x with no restrictions (that's the reason for the small "a"). If you graph this relation, you'll see that it fails the vertical-line test for a function. Choice B is the equivalent of y = ± x1/2 If you graph this relation without placing any restrictions on the domain or range, you'll see that the curve fails the vertical-line test for a function. Choice C is the equivalent of y = x/2 If you graph this relation with no restrictions on the domain or range, you'll get a straight line that passes the vertical-line test for a function. This choice C, therefore, is the correct answer. 6. When a mapping "morphs" one value of a given set to exactly one value of another set and vice-versa, and when all of the elements of both sets are accounted for, then that mapping is both an injection and a surjection. By definition, such a mapping constitutes a bijection. All three of the choices A, B, and C will work here, so the correct answer is D, "All of the above." 7. You can input any real number whatsoever (positive, negative, or 0) into the Arctangent function, and you'll always get a meaningful output value. The domain of the Arctangent function therefore extends over the entire set of real numbers. The correct choice is A. You can get some visual reinforcement for this fact by looking at Fig. 4-13, the graph of the Arctangent function. It's on page 95. 8. The range of possible output values for the Arccosecant function is limited to a set comprising two half-open intervals. One of the intervals is[-π/2,0) and the other is (0,π/2] That's the set of all reals between and including -π/2 and π/2, except 0. You can infer this fact by examining Fig. 4-14 on page 96. The correct choice is C. 9. The chapter text tells us that the domain of a mapping always forms a subset of the maximal domain, and the range of a mapping always forms a subset of the codomain. Therefore, statements A and B are both true. A mapping might have a range that encompasses all the real numbers, so statement D is true as well. The domain of a mapping might form a subset of the range, but not necessarily. Statement C does not hold true in general; as written, and interpreted literally, it's false. The correct choice is therefore C (because the question asks us which of the four statements is false). 10. In order to see which functions have inverse functions and which don't, you can graph all four functions. In each case, you'll get a straight line. The functions in choices B, C, and D show up as slanted lines in Cartesian coordinates. In all three of these situations, each value of the domain maps to exactly one value of the range, and vice-versa. If we transpose the domains and ranges of any of the three functions described in choices B, C, or D, we'll always get a one-to-one correspondence. However, the function described by choice A appears as a horizontal line in a Cartesian coordinate system where we let x represent the independent variable on the horizontal axis. Infinitely many values of the domain (the entire set of real numbers) map to the single value 3. If we transpose the domain and the range in an attempt to get an inverse for this function, we'll find that the single value 3 in the domain maps to infinitely many values in the range (all real numbers). Therefore, the function f (x) = 3 has no inverse function. The correct choice is A.