{{ item.displayTitle }}
navigate_next
No history yet!
Progress & Statistics equalizer Progress expand_more
Student
navigate_next
Teacher
navigate_next
{{ filterOption.label }}
{{ item.displayTitle }}
{{ item.subject.displayTitle }}
arrow_forward
{{ searchError }}
search
{{ courseTrack.displayTitle }}
{{ printedBook.courseTrack.name }} {{ printedBook.name }}

# Analyzing Functions in Context

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 $x$ and $y$ 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. $\begin{array}{c c c c c} \text{cost} & = & \text{unit price } & \times & \text{ weight}\\ y & = & 2.5 & \cdot & x \end{array}$

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

The table below shows the rate at which a $30$-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
$0$ $30$
$1$ $25$
$2$ $20$
$3$ $15$
Show Solution
Solution

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

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

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, $G$ is the dependent variable, giving us the function $G(m).$ By studying the table, we can find the function rule. Before the bathtub is drained, it contains $30$ gallons of water. Notice also that the amount of water decreases by $5$ gallons every minute. This yields the function rule $G(m) = 30 - 5m.$ Lastly, we have to figure out when the bathtub is completely empty. This happens when the function $G(m)$ reaches the value $0,$ since it represents the volume of water in the bathtub. By replacing $G(m)$ with $0$ in the function rule, we get an equation that we can solve for $m,$ giving us the desired time.

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

Thus, the bathtub is fully drained at $6$ minutes.

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 $\{0, 1, 2, 3, \ldots \}.$ 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 $\{\text{-} 0.5,0.5,0.7,0.9\},$ 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 $5$ or $6$ years old. Counting more precisely, he or she could be $16$ and a half, $10$ years and $3$ months, or even $25$ years and $172$ 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 $y$-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 $x$-intercept most often represents when or where the object touches the ground.

Method

### Function Values

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

that describes the cost, $C,$ in dollars for jumping in a bouncy castle for $t$ minutes. The statement $C(25) = 8$ could then be interpreted as $$jumping in the bouncy castle for $25$ minutes costs $8$ dollars." Note that this corresponds to the point $(25, 8)$ on the function graph.
fullscreen
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 $H(t)$ gives her height in feet above the surface of the water $t$ seconds since she's jumped.

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

Show Solution
Solution

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

This is the time at which her height above water is $0.$ 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 $y$-intercept is $H(0) = 6.$

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 $(2,4)$ on the graph tells us that $2$ seconds into her jump, her height above the water is $4$ feet.

fullscreen
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 $B(d)$ can be used to represent the balance, in dollars, for any day that is $d$ days following the first day of the month. Decide if the domain and range of $B$ are discrete or continuous. Then, propose a reasonable domain and range.

Show Solution
Solution

Since Ofelia checks her balance once a day, and here time is measured in whole days, $d$ must be a whole number. Thus, it is, in this context, a discrete quantity. The independent variable is $d,$ so the domain of $B$ is discrete. We also know that $d$ is the number of days after the first of the month. Therefore, it can vary between $0$ and $30,$ since no months have more than $31$ days. This gives us the reasonable domain $D : \{0, 1, \ldots, 30\}.$ 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 $5000$ dollars. This yields the proposed range $R : 0 \leq B(d) \leq 5000.$