Practice Exercises - Functions - AP Calculus Premium 2024
Practice Exercise
A1. If f(x) = x3 – 2x – 1, then f(–2) =
(A) –13
(B) –5
(C) –1
(D) 7
A2. The domain of is
(A) all x ≠ 1
(B) all x ≠ 1, –1
(C) all x ≠ –1
(D) all reals
A3. The domain of is
(A) all x ≠ 0, 1
(B) x ≤ 2, x ≠ 0, 1
(C) x ≥ 2
(D) x > 2
A4. If f(x) = x3 – 3x2 – 2x + 5 and g(x) = 2, then g (f(x)) =
(A) 2x3 – 6x2 – 2x + 10
(B) 2x2 – 6x + 1
(C) –3
(D) 2
A5. If f(x) = x3 – 3x2 – 2x + 5 and g(x) = 2, then f(g(x)) =
(A) 2x3 – 6x2 – 2x + 10
(B) 2x2 – 6x + 1
(C) –3
(D) 2
A6. If f(x) = x3 + Ax2 + Bx – 3 and if f(1) = 4 and f(–1) = –6, what is the value of 2A + B?
(A) 12
(B) 8
(C) 0
(D) –2
A7. Which of the following equations has a graph that is symmetric with respect to the origin?
(A)
(B) y = 2x4 + 1
(C) y = x3 + 2x
(D) y = x3 + 2
A8. Let g be a function defined for all reals. Which of the following conditions is not sufficient to guarantee that g has an inverse function?
(A) g(x) = ax + b, a ≠ 0
(B) g is strictly decreasing
(C) g is symmetric to the origin
(D) g is one-to-one
A9. Let y = f(x) = sin(arctan x). Then the range of f is
(A) {y | –1 < y < 1}
(B) {y | –1 ≤ y ≤ 1}
(C)
(D)
A10. Let g(x) = |cos x – 1|. The maximum value attained by g on the closed interval [0,2π] is for x equal to
(A) 0
(B)
(C) 2
(D) π
A11. Which of the following functions is not odd?
(A) f(x) = sin x
(B) f(x) = sin 2x
(C) f(x) = x3 + 1
(D)
A12. If the solutions to the equation f(x) = 0 are x = 1, –2, then f(2x) = 0 at x =
(A)
(B)
(C) 2 and –4
(D) –2 and 4
A13. The set of zeros of f(x) = x3 + 4x2 + 4x is
(A) {–2}
(B) {0,–2}
(C) {0,2}
(D) {2}
A14. The values of x for which the graphs of y = x + 2 and y2 = 4x intersect are
(A) –2 and 2
(B) –2
(C) 2
(D) no intersection
A15. The function whose graph is a reflection in the y-axis of the graph of f(x) = 1 – 3x is
(A) g(x) = 1 – 3–x
(B) g(x) = 3x – 1
(C) g(x) = log3 (x – 1)
(D) g(x) = log3 (1 – x)
A16. Let f(x) have an inverse function g(x). Then f(g(x)) =
(A) 1
(B) x
(C)
(D) f(x) ⋅ g(x)
A17. The function f(x) = 2x3 + x – 5 has exactly one real zero. It is between
(A) –1 and 0
(B) 0 and 1
(C) 1 and 2
(D) 2 and 3
A18. The period of is
A19. The range of y = f(x) = ln (cos x) is
(A) {y | –∞ < y ≤ 0}
(B) {y | 0 < y ≤ 1}
(C) {y | –1 < y < 1}
(D) {y | 0 ≤ y ≤ 1}
A20. If , then b =
A21. Let f–1 be the inverse function of f(x) = x3 + 2. Then f–1(x) =
A22. The set of x-intercepts of the graph of f(x) = x3 – 2x2 – x + 2 is
(A) {–1,1}
(B) {1,2}
(B) {–1,1,2}
(B) {–1,–2,2}
A23. If the domain of f is restricted to the open interval , then the range of f(x) = etan x is
(A) the set of all reals
(B) the set of positive reals
(C) the set of nonnegative reals
(D) {y | 0 < y ≤ 1}
A24.Which of the following is a reflection of the graph of y = f(x) in the x-axis?
(A) y = –f(x)
(B) y = f(–x)
(C) y = f(|x|)
(D) y = –f(–x)
A25. The smallest positive x for which the function is a maximum is
A26.
A27. If f–1(x) is the inverse of f(x) = 2e–x, then f–1(x) =
A28. Which of the following functions does not have an inverse function?
A29. Suppose that f(x) = ln x for all positive x and g(x) = 9 – x2 for all real x. The domain of f(g(x)) is
(A) {x | x ≤ 3}
(B) {x | |x| > 3}
(C) {x | |x| < 3}
(D) {x | 0 < x < 3}
A30.Suppose that f(x) = ln x for all positive x and g(x) = 9 – x2 for all real x. The range of y = f(g(x)) is
(A) {y | y > 0}
(B) {y | 0 < y ≤ ln 9}
(C) {y | y ≤ ln 9}
(D) {y | y < 0}
*A31. The curve defined parametrically by x(t) = t2 + 3 and y(t) = t2 + 4 is part of a(n)
(A) line
(B) circle
(C) parabola
(D) ellipse
*A32. Which equation includes the curve defined parametrically by x(t) = cos2(t) and y(t) = 2 sin(t)?
(A) x2 + y2 = 4
(B) 4x2 + y2 = 4
(C) 4x + y2 = 4
(D) x + 4y2 = 1
*A33. Find the smallest value of θ in the interval [0,2π] for which the rose r = 2 cos(5θ) passes through the origin.
*A34. For what value of θ in the interval [0,π] do the polar curves r = 3 and r = 2 + 2 cos θ intersect?
B1. The graph of the function f(x) = 2esin(x) – 3 crosses the x-axis once in the interval [0,1]. What is the x-coordinate of this x-intercept?
(A) 0.209
(B) 0.417
(C) 0.552
(D) 0.891
B2. Find the x-intercept of the graph of on the portion of the graph where f(x) is decreasing.
(A) –1.334
(B) –0.065
(C) –0.801
(D) 0.472
B3. You are given the function on the closed interval [–2,2]. Find all intervals where f(x) < 0.
(A) (–2,–1.421) and (0.305,1.407)
(B) (–1.421,0.305) only
(C) (–2,–1.421) only
(D) (0.305,1.407) only
B4. You are given the function f(x) = (4 – 2x – 2x2)cos(3x – 4) on the closed interval [–3,2]. How many times does f(x) cross the x-axis in the interval?
(A) four
(B) five
(C) six
(D) seven
*B5. On the interval [0,2π] there is one point on the curve r = θ – 2 cos θ whose x-coordinate is 2. Find the y-coordinate there.
(A) –4.594
(B) –3.764
(C) 1.979
(D) 5.201
B6. If f(x) = (1 + ex) then the domain of f–1(x) is
(A) (–∞,∞)
(B) (0,∞)
(C) (1,∞)
(D) (2,∞)