AP Precalculus Problems and Solutions

Problem 1 :

If q(x) = 5 – x² and p(q(x)) = (4 – x²)/ x² when x ≠ 0, then what is p(1/4) equal to? Show each of your steps in finding the answer. Explain each of the steps.

Solution :

Problem 2 :

Which of the following is equal to the above limit?

A)  1/6

B)  1/3

C)  3

D)  6

Solution :

Problem 3 :

Which of the following is equal to the above expression?

A)  2log₁₀3

B)  3log₁₀2

C)  5log₁₀2

D)  6log₁₀3

Solution :

Problem 4 :

Find the domain and range of f(x) = |x – 2|/(x + 5).

Solution :

Problem 5 :

The function h is given by h(x) = 8 ⋅ 2^x. For which of the following values of x, h(x) = 256?

A)  x = 2

B)  x = 5

C)  x = 8

D)  x = 16

Solution :

Problem 6 :

The function h is given by h(x) = log₃ x. Which of the following is equivalent to the expression

2 ⋅ h(w) + h(p),

where w and p are values in the domain of h?

A) log₃ (wp)²

B) (log₃ w)² ⋅ (log₃ p)

C) log₃ (w²p)

D) log₃ (2wp)

Solution :

Problem 7 :

Let x and y be positive constants. Which of the following is equivalent to 2ln x - 2 ln y ?

Solution :

Problem 8 :

The function f is given by f(x) = 2log₅ x. Which of the following describes f?

A) is an increasing function that increases at an increasing rate.

B) is an increasing function that increases at a decreasing rate.

C) is a decreasing function that decreases at an increasing rate.

D) is a decreasing function that decreases at a decreasing rate.

Solution :

Problem 9 :

The Richter scale is a numerical scale that uses base 10 logarithms for measuring an earthquake’s magnitude. The larger the number, the more intense the earthquake. As intensities increase multiplicatively by a factor of 10, the Richter scale increases additively by 1. Consider two earthquakes that occurred in the year 2022. An earthquake in Indonesia had a magnitude of 5.1, and an earthquake in Mexico had a magnitude of 2.5. Approximately how many times more intense was the Indonesia earthquake than the Mexico earthquake?

A)  2.6

B)  26

C)  100

D)  400

Solution :

Problem 10 :

The point P(3, 8) is on the graph of y = b^x, where b > 1.  The corresponding point P' on the graph of y + 3 = b^{x + 1} is

A)  (2, 11)

B)  (2, 5)

C)  (4, 11)

D)  (4, 5)

Solution :

Problem 11 :

The function y = f(x) has a domain of {x| 2 ≤ x ≤ 6, x ∈ R} and a range of {y| -4 ≤ y ≤ 8, y ∈ R}, The function undergoes the transformation y = -f(½x).

The domain and range of the transformed function are shown in row :

Solution :

Problem 12 :

The graph of y = f(x) is transformed into the graph of y = g(x), shown below.

An equation for g(x) in terms of f(x) is

A)  g(x) = f(2x) + 10

B)  g(x) = f(½x) + 10

C)  g(x) = 2f(2x)

D)  g(x) = 2f(½x)

Solution :

Problem 13 :

The loudness of a sound is related to the logarithm of the ratio of the measured intensity, I, to a reference intensity, I₀. The loudness, L, of a sound is measured in decibels, dB, and can be determined using the following formula.

L = 10log₁₀(I/I₀)

During an international soccer tournament in 2010, a noisemaker called the vuvuzela had a measured loudness of 127 dB at full volume.

If the intensity of the sound of the vuvuzela is 5 000 times greater than the intensity of the sound of a lawn mower, then the measured loudness of the lawn mower, to the nearest decibel, is

A)  3 dB

B)  37 dB

C)  90 dB

D)  123 dB

Solution :

Problem 14 :

Find the logarithm of 21952 to the base 2√7.

Solution :

Problem 15 :

If g(x) = 2x² and f(g(x)) = (3 – 4x²)/(8x² + 5), then what is f(1/2) equal to? 

Solution :

Problems 16-18 : Find the domain and range of each rational function.

Problem 16 :

Problem 17 :

Problem 18 :

Solutions (16-18) :

Problem 19 :

Which of the following lines are asymptotes of the graph of f(x)?

I.  x = ±3

II.  y = 5

III.  y = 5x

A)  I only

B)  II only

C)  I andf III only

D)  I and II only

Solution :

Problem 20 :

While a Ferris wheel is turning at a constant speed, the height above the ground, in feet, of a seat on the wheel is given by

h(t) = 29 + 25cos(π/3(t - 3))

where t is the time, in seconds, since the wheel strated turning. For how many seconds during the first complete turn of the wheel is the seat less than 29 feet above the ground?

A)  1.50

B)  3.00

C)  4.00

D)  6.00

Solution :

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