Analyzing Functions in Context

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Functions can be used to represent real-world situations involving two quantities. Rather than analyzing a given function in terms of inputs and outputs, the quantities that xx and yy each represent can be used.
Method

Representing a Function

Since a function is a type of relation, it can be represented in the same way. Commonly, functions are represented using tables, graphs, or rules.
Concept

Independent and Dependent Variables

For functions in context, the input is often referred to as the independent variable because it can be chosen arbitrarily from the domain. Conversely, the output is called the dependent variable, because its value depends on the value of the independent variable. For example, consider how the cost of fruit is determined by weight. cost=unit price × weighty=2.5x\begin{array}{c c c c c} \text{cost} & = & \text{unit price } & \times & \text{ weight}\\ y & = & 2.5 & \cdot & x \end{array}

If the unit price of clementines is $2.50/lb,\$2.50/\text{lb}, then the total cost of buying clementines depends on how many pounds are bought. Therefore, the cost yy of clementines is the dependent variable and the number of pounds xx is the independent variable.
Exercise

The table below shows the rate at which a 3030-gallon bathtub drains per minute for the first three minutes. Write the rule that represents the function. Then, use the rule to determine how many minutes it takes for the tub to empty.

Number of minutes draining Number of gallons remaining
00 3030
11 2525
22 2020
33 1515
Solution

We'll start by defining variables to represent the included quantities.

  • Let GG represent the number of gallons remaining in the bathtub.
  • Let mm represent the minutes since the tub began draining m.m.

We now have to determine which quantity depends on the other. Over time, we know the number of gallons changes based on how long the tub has been draining. However, time changes independently of the bathtub draining. Thus, GG is the dependent variable, giving us the function G(m). G(m). By studying the table, we can find the function rule. Before the bathtub is drained, it contains 3030 gallons of water. Notice also that the amount of water decreases by 55 gallons every minute. This yields the function rule G(m)=305m. G(m) = 30 - 5m. Lastly, we have to figure out when the bathtub is completely empty. This happens when the function G(m)G(m) reaches the value 0,0, since it represents the volume of water in the bathtub. By replacing G(m)G(m) with 00 in the function rule, we get an equation that we can solve for m,m, giving us the desired time.

G(m)=305mG(m) = 30 - 5m
0=305m{\color{#0000FF}{0}} = 30 - 5m
5m=305m = 30
m=6m = 6

Thus, the bathtub is fully drained at 66 minutes.

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Concept

Discrete Quantity

A discrete quantity is a quantity that can only take specific values. An example would be the number of times a person has gone skydiving. Notice that the value of that quantity can only be specific values like {0,1,2,3,}.\{0, 1, 2, 3, \ldots \}. If the independent variable of a function is a discrete quantity, it is said that the function has a discrete domain. Such a function can be recognized by its graph, which consists of any number of unconnected points.

Note that a discrete quantity doesn't have to be integers. For instance, if a quantity has the possible values {-0.5,0.5,0.7,0.9},\{\text{-} 0.5,0.5,0.7,0.9\}, it is discrete. In context, discrete quantities arise when working with things that can't be divided into infinitely many parts. This could for instance be how many children one has, or the number of apples purchases. These are most often restricted to whole numbers, but there are exceptions.
Concept

Continuous Quantity

A continuous quantity is a quantity that can take any value within one or several intervals. In other words, continuous quantities can be measured to any arbitrarily high degree of precision. Consider a person's age.

The age of a person is changing constantly, depending on how precisely it is measured. Counting only in years, a person could be 55 or 66 years old. Counting more precisely, he or she could be 1616 and a half, 1010 years and 33 months, or even 2525 years and 172172 days old.

Concept

Continuous Domain

If the input of a function is a continuous quantity, the function is said to have a continuous domain. The graph of such a function is typically a curve or a line.

Continuous quantities commonly include length, weight, volume, and difference in time, such as the volume of liquid in a drinking glass or the time that has passed since a person's last haircut.
Method

Interpreting Functions in Context

Functions in context can be interpreted in terms of domain and range, intercepts, and specific function values.

Method

Continuous and Discrete Range

Continuous and discrete range are defined similarly to continuous and discrete domain. If the dependent variable of a function is a discrete quantity, then it has a discrete range. If the dependent variable instead is a continuous quantity, then the function has a continuous range.

Method

Intercepts

The intercepts of a function in context are often of interest. For a function describing a cost, the yy-intercept typically represents a starting cost, which does not depend on how much the service is used. For a function describing the altitude of an object, the xx-intercept most often represents when or where the object touches the ground.

Method

Function Values

When a function value is given in the form f(a)=b,f(a) = b, for a function ff in context, there is always some interpretation that can be done. An example could be the function C(t)=3+0.2t, C(t) = 3 + 0.2t,

that describes the cost, C,C, in dollars for jumping in a bouncy castle for tt minutes. The statement C(25)=8C(25) = 8 could then be interpreted as ``jumping in the bouncy castle for 2525 minutes costs 88 dollars." Note that this corresponds to the point (25,8)(25, 8) on the function graph.
Exercise

Kenya is at a fictional lake, where gravity is significantly lower than normal. She decides to dive off a cliff into the lake. The function H(t)H(t) gives her height in feet above the surface of the water tt seconds since she's jumped.

Identify the independent and dependent variables. State and interpret the xx- and yy-intercepts. Interpret the point (2,4)(2,4) in terms of time and height.

Solution

The function H(t)H(t) implies that Kenya's height above the water is given as a function of time. Thus, tt is the independent variable, and HH is the dependent variable. From the graph we find that the xx-intercept is t=3.t = 3.

This is the time at which her height above water is 0.0. Thus, we can interpret it as the time it takes for her to reach the surface of the water — 3 seconds. Similarly, we find that the yy-intercept is H(0)=6.H(0) = 6.

This is her height above water at the moment she jumps. We can interpret this as the height of the cliff — 6 feet. Lastly, the point (2,4)(2,4) on the graph tells us that 22 seconds into her jump, her height above the water is 44 feet.

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Exercise

Ofelia uses one main bank account to manage her income and expenses. Once every day, she checks the balance of the account. The function B(d)B(d) can be used to represent the balance, in dollars, for any day that is dd days following the first day of the month. Decide if the domain and range of BB are discrete or continuous. Then, propose a reasonable domain and range.

Solution

Since Ofelia checks her balance once a day, and here time is measured in whole days, dd must be a whole number. Thus, it is, in this context, a discrete quantity. The independent variable is d,d, so the domain of BB is discrete. We also know that dd is the number of days after the first of the month. Therefore, it can vary between 00 and 30,30, since no months have more than 3131 days. This gives us the reasonable domain D:{0,1,,30}. D : \{0, 1, \ldots, 30\}. Although money is most often measured to the precision of hundredths, it could potentially take on any value in between. Therefore, it is reasonable for us to view her account balance as a continuous quantity, leading to the range being continuous. Assuming Ofelia doesn't overdraw her account, nor has her life savings in it, a reasonable maximum value could be 50005000 dollars. This yields the proposed range R:0B(d)5000. R : 0 \leq B(d) \leq 5000.

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Exercises

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