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,∞)