Domain Range Asymptotes Quiz: Domain, Range, and Vertical Asymptotes

The answer is True. The derivative of tan(theta) is sec squared theta, which equals 1 divided by cos squared theta. Since cos squared theta is always positive wherever it is defined, this derivative is always positive on the open intervals between asymptotes. A positive derivative confirms the function is strictly increasing, and the function is continuous throughout those same open intervals.

As theta approaches 3*pi/2 from the left, the angle is in the third quadrant where sin(theta) approaches -1 and cos(theta) approaches 0 from the negative side. Dividing -1 by a small negative number close to zero gives a large positive result. Therefore tan(theta) increases without bound toward positive infinity. Option A describes the behavior from the right side of this asymptote, not the left.

Between any two consecutive vertical asymptotes, tan(theta) rises continuously from negative infinity to positive infinity, passing through every real number exactly once per period. No real value is ever skipped or excluded from the output. Option B is wrong because 0 is included in the range. Option C describes the range of sine and cosine, not tangent. Option D incorrectly excludes all negative values.

The answer is False. The direction of the limit from the left depends on the specific asymptote. Approaching pi/2 from the left gives tan(theta) going to positive infinity, but approaching 3*pi/2 from the left, the angle is in the third quadrant where tangent is negative, so tan(theta) goes to negative infinity instead. The behavior alternates between positive and negative infinity at consecutive asymptotes.

The tangent function equals sin(theta) divided by cos(theta) and is undefined where cos(theta) = 0. These undefined points form an infinite set at theta = pi/2 + k*pi for every integer k. Option B incorrectly excludes the zeros of the tangent function itself. Option C only removes half the asymptote locations. Option D wrongly includes all the asymptote points without any exclusion.

The answer is True. On each open interval between consecutive vertical asymptotes, tan(theta) increases continuously from negative infinity to positive infinity. Because the function takes every real value within each such interval, no real number is excluded from the output. The range is therefore the entire set of real numbers, with no upper or lower bound and no gaps.

The domain of tan(theta) excludes all values where cos(theta) = 0, because division by zero is undefined. Cosine equals zero at theta = pi/2 + k*pi for every integer k, so those values must be removed from the real number line. Option A incorrectly excludes the zeros of sin(theta). Option B wrongly includes every real number. Option D misses half the asymptote locations.

Vertical asymptotes occur wherever the function is undefined, which happens when cos(theta) = 0. The cosine function equals zero at theta = pi/2 + k*pi for every integer k. Option A lists the zeros of the tangent itself, not its asymptotes. Option C only captures half the asymptote locations. Option D is a single point where the tangent is actually defined and equal to zero.

The tangent function is defined as sin(theta) divided by cos(theta). Division by zero is undefined in mathematics, so tan(theta) becomes undefined whenever the denominator cos(theta) equals zero. The other options — cos(theta) = 1, -1, or 1/2 — all represent nonzero denominator values, which produce perfectly valid outputs for the tangent function.

Vertical asymptotes occur exactly where cos(theta) = 0, which is at theta = pi/2 + kpi for every integer k. Consecutive asymptotes differ by pi, confirming they are spaced pi units apart. Option D is incorrect because the zeros of tan(theta) occur at theta = kpi, which are entirely different locations from the asymptotes. Options A, B, and C are all equivalent and accurate descriptions.

The answer is True. The tangent function equals zero when its numerator, sin(theta), equals zero. Sin(theta) = 0 exactly at theta = kpi for any integer k. At each of these values, cos(kpi) does not equal zero, so the tangent is defined and equals zero. Therefore every theta = k*pi is indeed a zero of the tangent function.

The tangent function is defined as sin(theta) divided by cos(theta), so it is undefined wherever cos(theta) = 0. Those values occur at theta = pi/2 + k*pi for every integer k. Options A and D both describe this same set of excluded points in two equivalent ways. Option B is false because it includes the asymptote locations. Option C incorrectly excludes the zeros of sin(theta) instead.

The answer is True. The interval (pi/2, 3*pi/2) spans one complete period of the tangent function. Within this interval, tan(theta) is continuous and strictly increasing, crossing zero exactly once at theta = pi. Since the function is monotone throughout and crosses zero only at pi, there is no other zero present in this interval.

The tangent function is positive when sine and cosine share the same sign. Within (-pi/2, pi/2), positive angles fall in the first quadrant where both sin and cos are positive, so tan is positive there. Negative angles fall in the fourth quadrant where sin is negative and cos is positive, giving a negative tangent. Therefore only pi/6 and pi/4 satisfy tan(theta) greater than 0.

Approaching pi/2 from the right places the angle in the second quadrant, where sin(theta) remains close to 1 but cos(theta) is a small negative number approaching 0 from below. Dividing a positive value near 1 by a small negative number produces a very large negative result, so tan(theta) decreases without bound toward negative infinity.

Vertical asymptotes for the tangent function occur at theta = pi/2 + kpi for any integer k. Substituting k = -2, -1, 0, and 1 gives -3pi/2, -pi/2, pi/2, and 3pi/2 respectively. All four values fall within [-2pi, 2*pi], and no additional asymptotes exist within that range, making option A the complete and correct list.

A single branch of the tangent graph lies strictly between two consecutive vertical asymptotes. The interval from -pi/2 to pi/2, not including the endpoints, fits this exactly because the function is continuous and strictly increasing throughout. Option D includes the endpoints pi/2 and 3*pi/2 where the function is undefined, so it cannot form a valid complete branch.

The tangent is undefined wherever cos(theta) = 0. On the interval [0, 2pi), cos(theta) = 0 at exactly two values: theta = pi/2 and theta = 3pi/2. At theta = 0, cos(0) = 1, and at theta = pi, cos(pi) = -1, so tan(theta) is defined at both those points and neither qualifies as an undefined location.

As theta approaches pi/2 from the left, the angle is in the first quadrant where both sin(theta) and cos(theta) are positive. sin(theta) approaches 1 while cos(theta) approaches 0 from the positive side. Dividing a value near 1 by a small positive number produces a very large positive result, so tan(theta) increases without bound toward positive infinity.

The tangent function repeats every pi radians, making pi its period. It is undefined wherever cos(theta) = 0, which produces vertical asymptotes at those points. The range of tan(theta) is all real numbers, which includes 0, so option C is false. Since tan(-theta) = -tan(theta), the function is odd. The domain excludes asymptote locations, so the fifth option stating the domain is all real theta is also false.