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 – 3^x is

(A) g(x) = 1 – 3^–x
(B) g(x) = 3^x – 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) = e^{tan 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) = 2e^sin(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 + e^x) then the domain of f^–1(x) is
(A) (–∞,∞)
(B) (0,∞)
(C) (1,∞)
(D) (2,∞)