PRACTICE EXERCISES Part A. No Calculator—Directions: Answer these questions without using your calculator. A1. If f(x) = x3 − 2x − 1, then f(−2) = (A) (B) (C) (D) −13 −5 −1 7 A2. The domain of f (x) = x−1 2 x +1 is (A) all x ≠ 1 (B) all x ≠ 1, −1 (C) all x ≠ −1 (D) all reals A3. The domain of g(x) = √ x−2 2 x −x 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) (B) (C) (D) 12 8 0 −2 A7. Which of the following equations has a graph that is symmetric with respect to the origin? (A) y = x−1 x (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) {y − (D) {y − π < y < 2 π ≦ y ≦ 2 π 2 π 2 } } 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 π (B) 2 (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) f (x) = x 2 x +1 A12. The roots of the equation f(x) = 0 are 1 and −2. The roots of f(2x) = 0 are (A) 12 and − 1 (B) − 1 2 and1 (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) 1 x (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 (B) f (x) = sin π 2 3 x is (A) 3 2 (C) 3 (D) 6 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} 2 3 A20. If b logb(3 ) = (A) 1 (B) 1 b 2 , then b = 9 2 (C) 3 (D) 9 A21. Let f−1 be the inverse function of f(x) = x3 + 2. Then f−1(x) = (A) 1 3 x −2 (B) (x − 2)3 (C) √ x+2 (D) √ x−2 3 3 A22. The set of x-intercepts of the graph of f(x) = x3 − 2x2 − x + 2 is (A) {−1,1} (B) {1,2} (C) {−1,1, 2} (D) {−1,−2,2} A23. If the domain of f is restricted to the open interval π π tan x is (A) the set (− , ), then the range of f(x) = e 2 2 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 f (x) = sin( x 3 ) − 1 is a maximum is (A) π 2 (B) π (C) π 3 2 (D) 3π A26. tan(arccos(− √2 2 )) = (A) −1 (B) (C) − √3 3 √3 3 (D) 1 A27. If f−1(x) is the inverse of f(x) = 2e−x, then f−1(x) = (A) ln( (B) ln( (C) ( (D) √ ln x 1 2 2 x x 2 ) ) ) ln x A28. Which of the following functions does not have an inverse function? (A) y = sin x(− (B) y = x3 + 2 (C) y = x x+1 2 π 2 ≦ x ≦ π 2 ) (D) y = ln (x − 2) (where x > 2) 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 (as in Question A29) 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} BC ONLY A31–A34 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. (A) 0 (B) (C) (D) π 20 π 10 π 5 A34. For what value of θ in the interval [0,π] do the polar curves r = 3 and r = 2 + 2 cos θ intersect? (A) π (B) π (C) π (D) 6 4 3 π 2 3 Part B. Calculator Active—Directions: Some of the following questions require the use of a graphing calculator. B1. The graph of the function f(x) = 2esin(x) − 3 crosses the x-axis once in the interval [0,1]. What is the xcoordinate 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 3 f (x) = √ sin(x) + 1 + x cos(x) − 3e + 6 of the graph where f(x) is decreasing. on the portion (A) –1.334 (B) –0.065 (C) –0.801 (D) 0.472 5 3 B3. You are given the function f(x) = ex – 2x + 2 – 7 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 BC ONLY 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 ycoordinate 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,∞)