CLEP Precalculus

If f(x) = x² − 2x and g(x) = 4x, which of the following is equal to fg(x) for x ≠ 0?

• A x² − 12
• B 4x² − 2
• C x²4 − 2x
• D x²4 − 12
• E x4 − 12

A ship, located at point S, is 20 miles due south of a buoy at point B. The ship sails at 16 miles per hour along a straight course that is 70° east of north. After 5 hours, what is the distance, in miles, between the ship and the buoy?

• A61.39
• B75.53
• C77.46
• D88.85
• E99.03

The shaded region in the figure is bounded by the y-axis, the line y = 2, and the curve y = x. If a point lies in the interior of the shaded region with coordinates (x, y), which of the following must be true?

I.  x < 4
II.  x < y
III.  x < y

• AI only
• BII only
• CI and II only
• DII and III only
• EI, II, and III

If log₄ x = 32, what is the value of x?

• A18
• B16
• C6
• D8
• E16

If a ≥ 0, then a³ · a⁵ =

• Aa
• Ba²
• Ca³
• Daa
• Ea²a

If w, x, and y are positive numbers, then 32wx²y2w³x² =

• A4x²5w
• B8yx²w
• C4yx·w
• D4yx²w
• E4yx³w

Let the function f be defined by: f(x) = ⎧ 1x−2  if x ≤ 1 x² − 4   if x > 1

What is the value of f(3) + f(−3)?

Which of the following is an equation of the line in the xy-plane that is parallel to the y-axis and passes through the point (5, −2)?

• Ax = 5
• By = 5
• Cx = −2
• Dy = −2
• Ex + y = 3

A balloon floats above and between two points A and B on the ground. The angles of elevation from each point to the balloon are 48° and 53°, respectively, and the line-of-sight distance to the balloon from A is 18 feet. What is the distance from A to B, to the nearest foot?

• A13 ft
• B14 ft
• C22 ft
• D24 ft
• EIt cannot be determined from the information given.

If f(x) = cos²x and g(x) = sin²x, which of the following defines the function h, where h(x) = 1 for all x?

• Af − g
• Bf + g
• Cfg
• Dfg
• Ef² + g²

Let f be the function defined by f(x) = 4x + 12. If g(x) = 12f(2x), which of the following is an expression for the function g?

• Ag(x) = 2x + 6
• Bg(x) = 2x + 12
• Cg(x) = 4x + 6
• Dg(x) = 4x + 12
• Eg(x) = 8x + 24

The surface area of a spherical balloon of radius r is given by f(r) = 4πr². As the balloon is inflated, the radius is given by g(t) = 3tt + 1, where t represents inflation time. Which of the following best describes the composite function f ∘ g?

• AThe surface area of the balloon as a function of inflation time
• BThe radius of the balloon as a function of inflation time
• CThe radius of the balloon as a function of the balloon's surface area
• DThe inflation time as a function of the balloon's surface area
• EThe inflation time as a function of the balloon's radius

The temperature, in °F, of a room t hours after midnight varies according to T(t) = A sinπt12 + B, where A and B are positive constants. The maximum temperature is 70°F and the greatest difference in temperature is 10°F. What is the value of B?

• A10
• B65
• C70
• D75
• E80

Manuel drove 100 miles total. The first part of the trip was at 60 mph; the rest at 40 mph. If n is the number of miles before heavy traffic, which function T gives the total hours driven?

• A T(n) = n60 + 100 − n40
• B T(n) = n40 + 100 − n60
• CT(n) = 60n + 40(100 − n)
• DT(n) = 40n + 60(100 − n)
• E T(n) = n(100 − n)(40)(60)

Which of the following could be a portion of the graph of the function f(x) = −1 + tan x?

[Five graph options displayed on original exam — the third option (graph shifted down 1 unit) is correct.]

• AGraph A — standard tan x (no shift)
• BGraph B — tan x shifted left
• CGraph C — tan x shifted down by 1 unit ✓
• DGraph D — reflected tan
• EGraph E — tan x shifted right

9y² − 4x² + 18y + 8x = 31 Which of the following is the graph in the xy-plane of the equation shown?

• AA circle
• BAn ellipse that is not a circle
• CA parabola
• DA hyperbola
• ETwo straight lines

The angle θ is in standard position. If cos θ > 0 and tan θ < 0, which of the following is true?

• A0 < θ < π2
• Bπ2 < θ < π
• Cπ < θ < 3π2
• D3π2 < θ < 2π
• E2π < θ < 9π4

The figure shows the graphs of the quadratic function f and the linear function g. What are all the x-intercepts of the graph of y = f(x) − g(x)?

• A1
• B1 and 3
• C1 and 5
• D−2 and 3
• E−2, 1, 3, and 5

Which of the following are trigonometric identities?

I.  sin xcos y = tanxy
II.  cos(−t) = −cos(t)
III.  sin(x²) = (sin x)²

• ANone
• BI only
• CII only
• DI and III
• EII and III

What is the least value of x that satisfies the equation 3x² + 13x − 10x + 7 = 0?

The figure shows a portion of the graph of a function g. If h is the function such that g(x) = h(x + 3) for all values of x, what is the y-intercept of the graph of h?

• A0
• B1
• C2
• D3
• E6

x + y = 1
2x − 3y = 3 If the ordered pair (x, y) is the solution to the system above, what is the value of x − y?

• A15
• B35
• C75
• D95
• E115

Which of the following is equal to (2x − 5)(3 + 4k)?
(Original image was low resolution; the selected answer was −5x + 1.)

• A−5x + 1 (selected on exam)
• B2x + 1
• C2x + 2
• D4x + 1
• E6x + 29

The graph of line n is shown above. Which of the following is an equation of the line that is parallel to n and contains the point (2, −3)?

• Ay + 3 = −52(x + 2)
• By + 3 = 52(x + 2)
• Cy + 3 = −54(x − 2)
• Dy + 3 = −52(x − 2)
• Ey + 3 = 52(x − 2)

If f(x) = 2cos(3x), then fπ18 =

• A0
• B32
• C1
• D3
• E22

Which of the following trigonometric functions has the greatest period?

• Acosx2
• B2 cos x
• Ccos(2x)
• D5 cos(3x)
• Ecos(4x)

When a certain company sells n items, its total sales revenue is modeled by R(n) = −(n − 30)² + 1,200. What is the average rate of change of the total sales revenue when the number of items sold increases from 20 to 28?

• A$14 per item
• B$42 per item
• C$85 per item
• D$1,142 per item
• E$2,284 per item

(See Q28 for same question with interval 20 to 26)

When a certain company sells n items, its total sales revenue is modeled by R(n) = −(n − 30)² + 1,200. What is the average rate of change of the total sales revenue when the number of items sold increases from 20 to 26?

• A$14 per item
• B$42 per item
• C$85 per item
• D$1,142 per item
• E$2,284 per item

f(x) = mx + 10     g(x) = kx − 10 For the functions above, m and k are constants where m < 0 and k > 0. If the graphs intersect at the point (6, 3), what are all values of x such that f(x) < g(x)?

• Ax < 3
• Bx < 6
• Cx > 3
• Dx > 6
• Ex > 10

If f(x) = x − 3 + 4, then f⁻¹, the inverse function of f, is defined by which of the following?

• Af⁻¹(x) = (x + 4)² − 3, for x ≥ 0
• Bf⁻¹(x) = (x + 3)² − 4, for x ≥ 0
• Cf⁻¹(x) = (x + 3)² − 4, for x ≥ 3
• Df⁻¹(x) = (x − 4)² − 3, for x ≥ 3
• Ef⁻¹(x) = (x − 4)² + 3, for x ≥ 4

If f(x) = x² − x, then f(x + 2) − f(x)2 =

• A1
• B4x
• C2x + 1
• D4x + 2
• Ex² − x

The figure shows the graph of a function g in the xy-plane. Which of the following could define g?

• Ag(x) = 2x + 3
• Bg(x) = 2 + 1x + 3
• Cg(x) = 2 + 1x − 3
• Dg(x) = 2 − 1x − 3
• Eg(x) = 2 + 1x

Which of the following is in the domain of the function f given by f(x) = log[(x + 3)(2x − 7)(3x − 25)]?

• A−4
• B3
• C4
• D5
• E6

What are the solutions to the equation sin³θ cosθ + sinθ cos³θ = 14 on the interval [0, π2]?

• Aθ = π12 only
• Bθ = π6 only
• Cθ = 5π12 only
• Dθ = π12 and θ = π6
• Eθ = π12 and θ = 5π12

N(t) = 4 + (t² − 2t)e^−0.5t, where 0 ≤ t ≤ 10

The function N models the squirrel population (in hundreds) in a wooded area over 10 years, where t = years after January 1, 1990. Between which years was the squirrel population at its maximum?

• A1990 and 1991
• B1991 and 1992
• C1994 and 1995
• D1995 and 1996
• E1999 and 2000

Let f be a one-to-one function with domain (−∞, ∞). The point (a, b) is on the graph of y = f(x), where a and b are nonzero real numbers. Which of the following points must be on the graph of the inverse function y = f⁻¹(x)?

• A(−a, −b)
• B(−b, −a)
• C(−1b, −1a)
• D(1a, 1b)
• E(b, a)

If sin θ = −513 and π < θ < 32π, which of the following has the least value?

• Atan θ
• Bcot θ
• Ccos θ
• Dsec θ
• Ecsc θ

In the xy-plane, the axis of symmetry of a parabola is the line x = 2. The parabola intersects the x-axis when x = −1. Which of the following could be an equation of the parabola?

• Ay = x² − x − 2
• By = x² − 2x − 3
• Cy = x² − 4x − 5
• Dy = x² + 6x + 5
• Ey = 2x² + x − 1

The range of the function f is {y : 0 ≤ y ≤ 2}. If g(x) = f(x − 5) + 10, what is the range of the function g?

• A{y : −5 ≤ y ≤ −3}
• B{y : 0 ≤ y ≤ 20}
• C{y : 5 ≤ y ≤ 7}
• D{y : 10 ≤ y ≤ 12}
• E{y : 10 ≤ y ≤ 20}

Which of the following is an equation of the line in the xy-plane that passes through the x- and y-intercepts of the graph of (x − 3)² + (y − 3)² = 9?

• Ay = x + 3
• By = −x + 3
• Cy = −x − 3
• Dy = −x + 6
• Ey = −x + 9

If −7x + 5 = 2x − b, which of the following expresses x in terms of b?

• Ax = −15(b + 5)
• Bx = −19(5 − b)
• Cx = 19(5 − b)
• Dx = 19(b + 5)
• Ex = 15(b + 5)

The cost of a container of liquid shaped as a rectangular box with interior dimensions 2.5 × 4 × 8 inches is $1.25. If cost is directly proportional to volume, what is the cost of a container with interior dimensions 4 × 10 × 12 inches?

• A$5.00
• B$6.00
• C$7.50
• D$11.25
• E$24.00

|x − m| < 2 If m > 0 in the inequality above, which of the following CANNOT be the value of x?

• A−2
• B0
• C2
• D2
• E4

f(x) = 10 sin(x²)x² + 1 There are two values of x at which the function f attains its absolute maximum value. What is the distance between the two corresponding maximum points on the graph of f in the xy-plane?

• A1.064
• B2.128
• C4.246
• D4.258
• E8.492

The figure shows the graph of the function f whose domain is {−2, −1, 0, 1, 2}. What is the set of all values of x such that |f(x)| < 2?

• A{0, 1}
• B{0, 1, 2}
• C{−1, 0, 1}
• D{−2, −1, 0, 1}
• E{−2, −1, 0, 1, 2}

If f(x) = sin(3x), then f(f(x)) =

• A9 sin²(x)
• Bsin²(3x)
• Csin²(9x²)
• Dsin(sin(3x))
• Esin(3 sin(3x))

What are all values of x for which 2 < 10^x < 5?

• A0 < x < log52
• Blog25 < x < 0
• Clog 2 < x < log 5
• D15 < x < 12
• E20 < x < 50

Let f be the function defined by f(x) = 9x − 4. If h ≠ 0, then f(2 + h) − f(2 − h)2h =

• A1
• B9
• C−4h
• D1 − 4h
• E9 − 4h