Calculus AB and Calculus BC
CHAPTER 1 Functions
Practice Exercises
Directions: Answer these questions without using your calculator.
1. If f (x) = x3 − 2x − 1, then f (−2) =
(A) −17
(B) −13
(C) −5
(D) −1
(E) 3
2. The domain of 
is
(A) all x ≠ 1
(B) all x ≠ 1, −1
(C) all x ≠ −1
(D) x 
1
(E) all reals
3. The domain of 
is
(A) all x ≠ 0, 1
(B) x 
2, x ≠ 0, 1
(C) x 2
(D) x 2
(E) x > 2
4. If f (x) = x3 − 3x2 − 2x + 5 and g(x) = 2, then g(f (x)) =
(A) 2x3 − 6x2 − 2x + 10
(B) 2x2 − 6 x + 1
(C) −6
(D) −3
(E) 2
5. With the functions and choices as in Question 4, which choice is correct for f (g(x))?
6. 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
(E) It cannot be determined from the given information.
7. 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
(E) 
8. 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 strictly increasing.
(E) g is one-to-one.
9. Let y = f (x) = sin (arctan x). Then the range of f is
(A) {y | 0 < y 1}
(B) {y | − 1 < y < 1}
(C) {y|−1 y 1}
(D) 
(E) 
10. Let g(x) = |cos x − 1|. The maximum value attained by g on the closed interval [0, 2π] is for x equal to
(A) −1
(B) 0
(C) 
(D) 2
(E) π
11. Which of the following functions is not odd?
(A) f (x) = sin x
(B) f (x) = sin 2x
(C) f (x) = x3 + 1
(D) 
(E) 
12. The roots of the equation f (x) = 0 are 1 and −2. The roots of f (2x) = 0 are
(A) 1 and −2
(B) 
(C) 
(D) 2 and −4
(E) −2 and 4
13. The set of zeros of f (x) = x3 + 4x2 + 4x is
(A) {−2}
(B) {0,−2}
(C) {0,2}
(D) {2}
(E) {2,−2}
14. 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) 0
(E) none of these
15. 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) = 1 + 3x
(C) g(x) = 3x − 1
(D) g(x) = log3 (x − 1)
(E) g(x) = log3 (1 − x)
16. Let f (x) have an inverse function g(x). Then f (g(x)) =
(A) 1
(B) x
(C) 
(D) f (x) · g(x)
(E) none of these
17. The function f (x) = 2x3 + x − 5 has exactly one real zero. It is between
(A) −2 and −1
(B) −1 and 0
(C) 0 and 1
(D) 1 and 2
(E) 2 and 3
18. The period of f (x) = 
is
(A) 
(B) 
(C) 
(D) 3
(E) 6
19. 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) 
(E) {y | 0 y 1}
20. If 
then b =
(A) 
(B) 
(C) 
(D) 3
(E) 9
21. Let f −1 be the inverse function of f (x) = x3 + 2. Then f −1(x) =
(A) 
(B) (x + 2)3
(C) (x − 2)3
(D) 
(E) 
22. The set of x-intercepts of the graph of f (x) = x3 − 2x2 − x + 2 is
(A) {1}
(B) {−1,1}
(C) {1,2}
(D) {−1,1,2}
(E) {−1,−2,2}
23. 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}
(E) none of these
24. 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|)
(E) y = −f (−x)
25. The smallest positive x for which the function 
is a maximum is
(A)
(B) π
(C) 
(D) 3π
(E) 6π
26. 
(A) −1
(B) 
(C) 
(D) 
(E) 1
27. If f −1(x) is the inverse of f (x) = 2e−x, then f −1(x) =
(A) 
(B) 
(C) 
(D) 
(E) ln (2 − x)
28. Which of the following functions does not have an inverse function?
(A) 
(B) y = x3 + 2
(C) 
(D) 
(E) y = ln (x − 2) (where x >2)
29. 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 | |x| < 3}
(E) {x | 0 < x < 3}
30. Suppose (as in Question 29) 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}
(E) none of these
31. 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
(E) hyperbola
BC ONLY
32. Which equation includes the curve defined parametrically by x(t) = cos2 (t) and y(t) = 2 sin (t)?
(A) x2 + y2 = 4
(B) x2 + y2 = 1
(C) 4x2 + y2 = 4
(D) 4x + y2 = 4
(E) x + 4y2 = 1
BC ONLY
33. Find the smallest value of 
in the interval [0,2π] for which the rose r = 2 cos(5) passes through the origin.
(A) 0
(B) 
(C) 
(D) 
(E) 
BC ONLY
34. For what value of in the interval [0,π] do the polar curves r = 3 and r = 2 + 2 cos intersect?
(A) 
(B) 
(C) 
(D) 
(E) 
BC ONLY
35. On the interval [0,2π] there is one point on the curve r = − 2 cos whose x-coordinate is 2. Find the y-coordinate there.
(A) −4.594
(B) −3.764
(C) 1.979
(D) 4.263
(E) 5.201
BC ONLY