AP Calculus BC Practice Quiz: Confirming Continuity over an Interval
Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026
Test your understanding with short quizzes. This quiz has 9 questions to check your progress.
Question 1 of 9
All Questions (9)
A) (-∞, ∞)
B) (0, ∞)
C) (-5, 7)
D) The function is not continuous anywhere.
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 is all real numbers, (-∞, ∞). Therefore, the function is continuous on (-∞, ∞). [cite: 1396]
A) (-∞, ∞)
B) (-∞, 4) U (4, ∞)
C) (-∞, -2) U (-2, ∞)
D) [4, ∞)
Correct Answer: B
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 values that make the denominator zero. Here, the denominator is zero when x - 4 = 0, or x = 4. Therefore, the function is continuous on its domain, which is all real numbers except x = 4, expressed as (-∞, 4) U (4, ∞). [cite: 1391, 1396]
A) The function must be continuous only at x = a and x = b.
B) The function's graph must be a straight line over the interval.
C) The function must be continuous at each point in the interval [a, b].
D) The function must be a polynomial.
Correct Answer: C
The definition of continuity on an interval states that a function is continuous on an interval if it is continuous at each point within that interval. [cite: 1394]
A) (-∞, ∞)
B) (-∞, 5)
C) (5, ∞)
D) [5, ∞)
Correct Answer: C
The function h(x) is a logarithmic function. Logarithmic functions are continuous at all points in their domains. The domain of ln(u) requires the argument u to be positive. For h(x) = ln(x - 5), we must have x - 5 > 0, which means x > 5. Therefore, the interval of continuity is (5, ∞). [cite: 1389, 1396]
A) f(x) = tan(x)
B) f(x) = 1/x
C) f(x) = sin(x)
D) f(x) = ln(x)
Correct Answer: C
According to the provided content, trigonometric functions are continuous on their domains. The domain of sin(x) is all real numbers, (-∞, ∞). The function tan(x) has discontinuities where cos(x) = 0. The function 1/x is a rational function with a discontinuity at x = 0. The function ln(x) is a logarithmic function with a domain of (0, ∞). Only sin(x) is continuous on (-∞, ∞). [cite: 1396]
A) (-∞, ∞)
B) (-3, 3)
C) [3, ∞)
D) [-3, 3]
Correct Answer: D
The function f(x) is a power function (specifically, a square root). Power functions are continuous on their domains. For the square root function, the expression inside the radical must be non-negative. Therefore, we must have 9 - x² ≥ 0. This inequality holds when -3 ≤ x ≤ 3. The interval of continuity is [-3, 3]. [cite: 1390, 1396]
A) (0, ∞)
B) (-∞, 0)
C) (-∞, ∞)
D) (1, ∞)
Correct Answer: C
The function k(x) is an exponential function. Exponential functions are continuous at all points in their domains. The domain of e^(2x) is all real numbers. Therefore, the function k(x) is continuous on the interval (-∞, ∞). [cite: 1396]
A) x = -1 only
B) x = -2 and x = 3
C) x = 2 and x = -3
D) x = -1, x = -2, and x = 3
Correct Answer: B
A rational function is continuous everywhere in its domain, and it is discontinuous where its denominator is zero. To find the points of discontinuity, we set the denominator equal to zero: x² - x - 6 = 0. Factoring the quadratic gives (x - 3)(x + 2) = 0. The solutions are x = 3 and x = -2. These are the x-values where the function is discontinuous. [cite: 1391, 1396]
A) Rational functions
B) Logarithmic functions
C) Polynomial functions
D) Power functions with non-integer exponents
Correct Answer: C
Polynomial functions have a domain of all real numbers, (-∞, ∞), and are continuous on their entire domain. Rational functions can have discontinuities where the denominator is zero. Logarithmic functions have restricted domains (e.g., ln(x) is only defined for x > 0). Power functions like √(x) also have restricted domains. Therefore, only polynomial functions are guaranteed to be continuous on (-∞, ∞). [cite: 1396]