Function Frontiers

Domain & Range Word Problems

Student:

Date:

Mission Briefing: Analyze each real-world scenario to determine the most reasonable domain or range. Pay close attention to whether the data is continuous or discrete, and look for physical constraints that define the boundaries of the function.

1

A local delivery service charges a flat fee of $12.00 plus $2.50 for every mile driven. The driver travels between 5 and 20 miles for each delivery. What is the range of this cost function?

A

\( 5 \le x \le 20 \)

B

\( 12 \le y \le 62 \)

C

\( 24.50 \le y \le 62 \)

D

\( 0 \le y \le 62 \)

2

A model rocket is launched from the ground. Its height in feet after \( t \) seconds is modeled by \( h(t) = -16t^2 + 64t \). The rocket hits the ground after 4 seconds. What is the domain of this height function?

A

\( 0 \le t \le 4 \)

B

\( 0 \le h \le 64 \)

C

\( \{0, 1, 2, 3, 4\} \)

D

\( t \ge 0 \)

3

A population of 50 bacteria doubles every hour for a 24-hour observation period. The function \( P(t) = 50(2)^t \) represents the population after \( t \) hours. Which statement best describes the range of this function?

A

The set of all integers from 50 to \( 50(2)^{24} \).

B

The set of all real numbers such that \( 0 \le y \le 24 \).

C

The set of all integers \( y \) such that \( y = 50(2)^t \) for \( t \in \{0, 1, 2, \dots, 24\} \).

D

The set of all real numbers such that \( 50 \le y \le 838,860,800 \).

4

A student earns $15 for every lawn mowed. During the summer, the student has a goal to mow at least 1 but no more than 10 lawns per week. What is the domain of the weekly earnings function?

A

\( \{15, 30, 45, \dots, 150\} \)

B

\( 1 \le x \le 10 \)

C

\( \{1, 2, 3, \dots, 10\} \)

D

\( \{0, 1, 2, \dots, 10\} \)

5

A rectangular garden is built against a wall using 40 feet of fencing for the other three sides. The area of the garden is given by \( A(w) = w(40 - 2w) \), where \( w \) is the width. What is the domain of this area function?

A

\( 0 < w < 20 \)

B

\( 0 < w < 40 \)

C

\( 0 < A \le 200 \)

D

\( 0 \le w \le 40 \)

6

A computer worth $1,200 depreciates by 20% each year. The value \( V(t) = 1200(0.80)^t \) represents the value after \( t \) years. What is the range for the first 5 years of ownership?

A

\( 0 \le V \le 1200 \)

B

\( 393.22 \le V \le 1200 \)

C

\( 0 \le t \le 5 \)

D

\( 240 \le V \le 1200 \)

7

A water tank holds 5,000 gallons of water and is being drained at a constant rate of 250 gallons per hour. Which of the following describes the domain of the amount of water remaining in the tank?

A

\( 0 \le x \le 20 \)

B

\( 0 \le y \le 5000 \)

C

\( x \ge 0 \)

D

\( 0 \le x \le 5000 \)

8

A company's profit \( P \) in thousands of dollars is modeled by \( P(x) = -x^2 + 10x - 16 \), where \( x \) is the price per unit in dollars. The company only considers prices that yield a profit. What is the domain?

A

\( 2 \le x \le 8 \)

B

\( 0 \le x \le 10 \)

C

\( 0 \le P \le 9 \)

D

\( 2 < x < 8 \)

9

An experiment starts with 2,000 mg of a substance that decays by 50% every day. The scientist measures the substance for exactly one week. What is the range of the remaining mass?

A

\( \{15.625, \dots, 2000\} \)

B

\( 0 \le y \le 7 \)

C

\( 15.625 \le y \le 2000 \)

D

\( 0 \le y \le 2000 \)

10

A rental shop charges $20 for a surfboard rental plus $5 per hour. A customer rents a surfboard for at least 2 hours but no more than 8 hours. What is the range of the rental cost?

A

\( 2 \le x \le 8 \)

B

\( 30 \le y \le 60 \)

C

\( 20 \le y \le 60 \)

D

\( 30 \le y \le 40 \)

11

A ball is thrown from a height of 6 feet. Its height \( H \) after \( t \) seconds is given by \( H(t) = -16t^2 + 20t + 6 \). The ball reaches its maximum height at \( t = 0.625 \) seconds. What is the range?

A

\( 0 \le H \le 6 \)

B

\( 0 \le H \le 12.25 \)

C

\( 0 \le t \le 1.5 \)

D

\( 6 \le H \le 12.25 \)

12

A computer game increases in complexity. Level 1 starts with 3 enemies, and the number of enemies doubles each level. The game ends after level 6. What is the domain of this level function?

A

\( \{1, 2, 3, 4, 5, 6\} \)

B

\( 1 \le x \le 6 \)

C

\( \{3, 6, 12, 24, 48, 96\} \)

D

\( x \ge 1 \)

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