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AP Precalculus -1.14 Function Models: Construction and Application- MCQ Exam Style Questions - Effective Fall 2023

AP Precalculus -1.14 Function Models: Construction and Application- MCQ Exam Style Questions – Effective Fall 2023

AP Precalculus -1.14 Function Models: Construction and Application- MCQ Exam Style Questions – AP Precalculus- per latest AP Precalculus Syllabus.

AP Precalculus – MCQ Exam Style Questions- All Topics

Question 

 
 
 
 
 
 
 
 
 
 
 
 
In 2016, the cost to mail a package was $2.54 for up to 3 ounces, plus an additional cost of $0.20 for each additional ounce or portion of an ounce less than a full ounce. A portion of the graph of this relationship is given with cost, in dollars, as a function of ounces. Which of the following describes the restrictions on the range for such a function?
(A) The range is positive real numbers.
(B) The range is positive integers.
(C) The range is values of the form \( 2.54 + 0.2x \), where \( x \) is a nonnegative real number.
(D) The range is values of the form \( 2.54 + 0.2x \), where \( x \) is a nonnegative integer.
▶️ Answer/Explanation
Detailed solution

Cost structure: $2.54 for up to 3 ounces, then $0.20 for each additional ounce or portion of an ounce less than a full ounce. This means weight exceeding 3 ounces is rounded up to the next integer ounce for pricing beyond 3 ounces.

Let \( w \) = weight in ounces, \( w > 3 \). Additional ounces counted = \( \lceil w – 3 \rceil \) (ceiling function). Let \( x = \lceil w – 3 \rceil \), a nonnegative integer.

Cost = \( 2.54 + 0.20x \), \( x \) ∈ {0, 1, 2, …}. Thus the range is a set of discrete values of that form, not all real numbers, because \( x \) is integer.

✅ Answer: (D)

Question 

Jordan’s cell phone plan includes 5 gigabytes (GB) of data per month and has a monthly cost of $79.95. If Jordan uses more than 5 GB of data within the month, there is a charge of $10 per additional gigabyte of data used. Function \( C \) is used to model Jordan’s monthly cell phone bill, where \( d \) is the number of gigabytes of data used and \( C(d) \) is the cost in dollars. Which of the following defines function \( C \)?
(A) \( C(d) = \begin{cases} 79.95 & \text{for } 0 \leq d \leq 5 \\ 79.95 + 10(d – 5) & \text{for } d > 5 \end{cases} \)
(B) \( C(d) = \begin{cases} 79.95 & \text{for } 0 \leq d \leq 5 \\ 79.95 + 10d & \text{for } d > 5 \end{cases} \)
(C) \( C(d) = \begin{cases} 79.95 & \text{for } 0 \leq d \leq 5 \\ 79.95d + 10 & \text{for } d > 5 \end{cases} \)
(D) \( C(d) = 79.95 + 10d \)
▶️ Answer/Explanation
Detailed solution

For \( 0 \leq d \leq 5 \), cost = base price $79.95.
For \( d > 5 \), extra data = \( d – 5 \) GB, charged at $10 per extra GB, so cost = \( 79.95 + 10(d – 5) \).
The piecewise function matches this.
✅ Answer: (A)

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