AP Calculus AB Practice Quiz: Confirming Continuity over an Interval

Correct Answer: A

The function f(x) is a polynomial. According to the provided content, polynomial functions are continuous on all points in their domains. The domain of any polynomial function is all real numbers, (-∞, ∞). Therefore, the function is continuous over the interval (-∞, ∞). [cite: 1396, 1394]

Correct Answer: C

The function g(x) is a rational function. Rational functions are continuous at all points in their domains. The domain of a rational function excludes any values of x that make the denominator zero. For g(x), the denominator is zero when x - 4 = 0, which means x = 4. Therefore, the domain is all real numbers except 4. The function is continuous on the intervals (-∞, 4) and (4, ∞). [cite: 1396, 1394]

Correct Answer: D

The function h(x) is a logarithmic function. Logarithmic functions are continuous at all points in their domains. The domain of a logarithmic function ln(u) requires its argument u to be strictly positive. For h(x) = ln(x - 2), we must have x - 2 > 0, which implies x > 2. Therefore, the domain, and the interval of continuity, is (2, ∞). [cite: 1396, 1394]

Correct Answer: B

The function f(x) is a power function (specifically, a square root function). Power functions are continuous at all points in their domains. The domain of a square root function requires its argument to be non-negative. For f(x) = √(x + 3), we must have x + 3 ≥ 0, which implies x ≥ -3. Therefore, the domain, and the interval of continuity, is [-3, ∞). [cite: 1396, 1394]

Correct Answer: C

The provided content explicitly states that 'A function is continuous on an interval if the function is continuous at each point in the interval.' Options A and B are necessary conditions for continuity at a point, but not sufficient. Option D lists types of functions that are continuous on their domains but does not define continuity on an interval. [cite: 1394]

Correct Answer: C

The function k(x) is a rational function, which is continuous on its domain. The domain excludes values where the denominator is zero. We set x² - 9 = 0, which factors to (x - 3)(x + 3) = 0. The solutions are x = 3 and x = -3. Therefore, the function is continuous everywhere except at x = -3 and x = 3. The intervals of continuity are (-∞, -3), (-3, 3), and (3, ∞). [cite: 1396, 1394]

Correct Answer: B

Trigonometric functions are continuous on their domains. The function tan(x) is defined as sin(x)/cos(x). Its domain excludes values where cos(x) = 0. This occurs at x = π/2 + nπ for any integer n. The interval (-π/2, π/2) does not contain any of these points of discontinuity. The interval (0, π) contains the discontinuity at x = π/2. The interval [0, π] also contains this discontinuity. The function is not continuous on (-∞, ∞) due to its periodic discontinuities. [cite: 1396, 1394]

Correct Answer: B

For a function to be continuous, it must be defined on its domain. The function h(x) involves a power function (√x) and is a rational expression. We must consider the domains of all parts. For √x, we need x ≥ 0. For the denominator √x to be non-zero, we need x ≠ 0. Combining these two conditions (x ≥ 0 and x ≠ 0), the domain is x > 0. Since rational and power functions are continuous on their domains, h(x) is continuous on (0, ∞). [cite: 1396, 1394]

Correct Answer: B

This function is continuous on its domain. To find the domain, we must satisfy two conditions. First, for the logarithmic function ln(x+2) to be defined, its argument must be positive: x + 2 > 0, which means x > -2. Second, for the rational function, the denominator cannot be zero: ln(x+2) ≠ 0. This occurs when x+2 ≠ 1, so x ≠ -1. Combining both conditions (x > -2 and x ≠ -1), the domain is (-2, -1) U (-1, ∞). This is the set of intervals where the function is continuous. [cite: 1396, 1394]