Question

domain & range homework: due next class
#1 - 4. decide a reasonable domain and range for the following situations. then determine whether the related functions are continuous or discrete.

1. joe had a summer job that pays $7.00 an hour and he worked between 15 and 35 hours every week. his weekly salary can be modeled by the equation: s = 7h, where s is his weekly salary and h is the number of hours he worked in a week.
2. the surface area of a cube can be found using the following formula: a = 6s², where a represents the surface area of the cube and s represents the length of one edge. your geometry teacher wants you to draw a cube that has a length between 1 and 5 inches.
3. you conduct an experiment on the speed of sound waves in dry air at 86°. you record your data in a table (pictured below).
4. a 20 - gallon bathtub is draining at a rate of 2.5 gallons per minute. the number of gallons g remaining is a function of the number of minutes m.
5. the table below represents the height of a building as a function of the number of stories the building has.
6. the drama club is planning a trip to an amusement park. they are taking a bus which holds 32 people. it will cost $25 for parking and tickets to enter the park and $22.50 per person. the equation that models this situation is c(n)=22.5n + 25, where c represents the cost for the group to go to the park and represents the number of people who go on this excursion.

Explanation:

1. For Joe's salary situation ($S = 7h$):

Step1: Determine the domain

Joe worked between 15 and 35 hours every week. So the domain (set of input values, which is the number of hours $h$) is $15\leq h\leq35$. Since the number of hours can take on any value within this interval (e.g., 15.5 hours), it is a continuous variable.

Step2: Determine the range

We substitute the endpoints of the domain into the equation $S = 7h$. When $h = 15$, $S=7\times15 = 105$. When $h = 35$, $S=7\times35=245$. So the range (set of output values, which is the salary $S$) is $105\leq S\leq245$. The function is continuous because the domain is continuous.

2. For the surface - area of a cube ($A = 6s^{2}$):

Step1: Determine the domain

The length of one edge $s$ is between 1 and 5 inches. So the domain is $1\leq s\leq5$. Since $s$ can take on any value within this interval (e.g., 2.3 inches), it is a continuous variable.

Step2: Determine the range

We substitute the endpoints of the domain into the equation $A = 6s^{2}$. When $s = 1$, $A=6\times1^{2}=6$. When $s = 5$, $A=6\times5^{2}=6\times25 = 150$. So the range is $6\leq A\leq150$. The function is continuous because the domain is continuous.

3. For the speed - of - sound experiment:

The domain is the set of input values (time in seconds in this case, as it is the independent variable). From the table, the domain is $\{2,4,6,8,10\}$. Since these are discrete values (only these specific time points are recorded), it is a discrete function. The range is the set of output values (speed values in the table), which is $\{0.834,0.868,1.302,1.736,2.170\}$.

4. For the draining bathtub ($g=20 - 2.5m$):

Step1: Determine the domain

The number of gallons $g$ must be non - negative. We set $g\geq0$, so $20 - 2.5m\geq0$. Solving for $m$ gives $2.5m\leq20$, or $m\leq8$. Also, $m\geq0$ (we can't have negative time). So the domain is $0\leq m\leq8$. Since $m$ can take on any value within this interval (e.g., 3.5 minutes), it is a continuous variable.

Step2: Determine the range

When $m = 0$, $g = 20$. When $m = 8$, $g=20-2.5\times8=0$. So the range is $0\leq g\leq20$. The function is continuous because the domain is continuous.

5. For the height of the building:

The domain is the set of the number of stories $\{1,2,3,4,5\}$. Since these are discrete values (you can't have a fraction of a story), it is a discrete function. The range is the set of building heights $\{12,24,36,48,60\}$.

6. For the drama - club trip ($c = 25+22.5n$):

Step1: Determine the domain

The number of people $n$ must be a non - negative integer. Also, since the bus holds 32 people, $0\leq n\leq32$ and $n\in\mathbb{Z}$ (where $\mathbb{Z}$ is the set of integers). So the domain is a set of discrete values.

Step2: Determine the range

When $n = 0$, $c = 25$. When $n = 32$, $c=25+22.5\times32=25 + 720=745$. So the range is $\{c\in\mathbb{R}:25\leq c\leq745,c = 25 + 22.5n,n\in\mathbb{Z},0\leq n\leq32\}$. The function is discrete because the domain is discrete.

Answer:

1. Domain: $15\leq h\leq35$, Range: $105\leq S\leq245$, Continuous.
2. Domain: $1\leq s\leq5$, Range: $6\leq A\leq150$, Continuous.
3. Domain: $\{2,4,6,8,10\}$, Range: $\{0.834,0.868,1.302,1.736,2.170\}$, Discrete.
4. Domain: $0\leq m\leq8$, Range: $0\leq g\leq20$, Continuous.
5. Domain: $\{1,2,3,4,5\}$, Range: $\{12,24,36,48,60\}$, Discrete.
6. Domain: $\{n\in\mathbb{Z}:0\leq n\leq32\}$, Range: $\{c\in\mathbb{R}:25\leq c\leq745,c = 25 + 22.5n,n\in\mathbb{Z},0\leq n\leq32\}$, Discrete.