Unit 5 Functions — Unit Plan

Here are graphs of two functions, each representing the cost of riding in a taxi from two companies—Friendly Rides and Great Cabs.

For each taxi, the cost of a ride is a function of the distance traveled. The input is distance in miles, and the output is cost in dollars.

2 graphs. Horizontal axis, 0 to 7, distance, miles. Vertical axis, 0 to 16 by 4’s, cost, dollars. Dotted graph starts flat, then increases. Passes through 2 comma 4 point 2 5. Solid graph starts flat then increases. Passes through 2 comma 5 point 7. They intersect near 4 point 5 comma 10.

• The point $(2,5.70)$ on one graph tells us the cost of riding a Friendly Rides taxi for 2 miles.
• The point $(2, 4.25)$ on the other graph tells us the cost of riding a Great Cabs taxi for 2 miles.

We can convey the same information much more efficiently by naming each function and using function notation to specify the input and the output.

• Let’s name the function for Friendly Rides function $f$.
• Let's name the function for Great Cabs function $g$.
• To refer to the cost of riding each taxi for 2 miles, we can write $f(2)$ and $g(2)$.
• To say that a 2-mile trip with Friendly Rides will cost $5.70, we can write $f(2)=5.70$.
• To say that a 2-mile trip with Great Cabs will cost $4.25, we can write $g(2)=4.25$.

In general, function notation has this form:

It is read “$f$ of $x$” and can be interpreted to mean that $f(x)$ is the output of a function $f$ when $x$ is the input.

The function notation is a concise way to refer to a function and describe its input and output, which can be very useful. Throughout this unit and the course, we will use function notation to talk about functions.

What does a statement like $p(3)=12$ mean?

On its own, $p(3)=12$ only tells us that when $p$ takes 3 as its input, its output is 12.

If we know what quantities the input and output represent, however, we can learn much more about the situation that the function represents.

• If function $p$ gives the perimeter of a square whose side length is $x$ and both measurements are in inches, then we can interpret $p(3)=12$ to mean “a square whose side length is 3 inches has a perimeter of 12 inches.”

  We can also interpret statements like $p(x)=32$ to mean “a square with side length $x$ has a perimeter of 32 inches,” which then allows us to reason that $x$ must be 8 inches and to write $p(8)=32$.

• If function $p$ gives the number of blog subscribers, in thousands, $x$ months after a blogger started publishing online, then $p(3)=12$ means “3 months after a blogger starts publishing online, the blog has 12,000 subscribers.”

It is important to pay attention to the units of measurement when analyzing a function. Otherwise, we might mistake what is happening in the situation. If we miss that $p(x)$ is measured in thousands, we might misinterpret $p(x)=36$ to mean “there are 36 blog subscribers after $x$ months,” while it actually means “there are 36,000 subscribers after $x$ months.”

A graph of a function can likewise help us interpret statements in function notation.

Function $f$ gives the depth, in inches, of water in a tub as a function of time, $t$, in minutes, since the tub started being drained.

Here is a graph of $f$.

Each point on the graph has the coordinates $(t, f(t))$, where the first value is the input of the function and the second value is the output.

• $f(2)$ represents the depth of water 2 minutes after the tub started being drained. The graph passes through $(2,5)$, so the depth of water is 5 inches when $t= 2$. The equation $f(2)=5$ captures this information.

• $f(0)$ gives the depth of the water when the draining began, when $t=0$. The graph shows the depth of water to be 6 inches at that time, so we can write $f(0)=6$.

• $f(t)= 3$ tells us that $t$ minutes after the tub started draining, the depth of the water is 3 inches. The graph shows that this happens when $t$ is 6.

Some functions are defined by rules that specify how to compute the output from the input. These rules can be verbal descriptions or expressions and equations. For example:

Some functions are defined by rules that relate two quantities in a situation. These functions can also be expressed algebraically with function notation.

Suppose function $c$ gives the cost of buying $n$ pounds of apples at $1.49 per pound. We can write the rule $c(n) = 1.49n$ to define function $c$.

To see how the cost changes when $n$ changes, we can create a table of values.

pounds of apples, $n$	cost in dollars, $c(n)$
0	0
1	1.49
2	2.98
3	4.47
$n$	$1.49n$

Plotting the pairs of values in the table gives us a graphical representation of $c$.

The domain of a function is the set of all possible input values. Depending on the situation represented, a function may take all numbers as its input or only a limited set of numbers.

• Function $A$ gives the area of a square, in square centimeters, as a function of its side length, $s$, in centimeters.

  • The input of $A$ can be 0 or any positive number, such as 4, 7.5, or $\frac{19}{3}$. It cannot include negative numbers because lengths cannot be negative.
  • The domain of $A$ includes 0 and all positive numbers (or $s \geq 0$).

• Function $q$ gives the number of buses needed for a school field trip as a function of the number of people, $n$, going on the trip.

  • The input of $q$ can be 0 or positive whole numbers because a negative or fractional number of people doesn’t make sense.
  • The domain of $q$ includes 0 and all positive whole numbers. If the number of people at a school is 120, then the domain is limited to all non-negative whole numbers up to 120 (or $0 \leq n \leq 120$).

• Function $v$ gives the total number of visitors to a theme park as a function of days, $d$, since a new attraction opened to the public.

  • The input of $v$ can be positive or negative. A positive input means days since the attraction opened, and a negative input means days before the attraction opened.
  • The input can also be whole numbers or fractional. The statement $v(17.5)$ refers to 17.5 days after the attraction opened.
  • The domain of $v$ includes all numbers. If the theme park had opened exactly one year before the new attraction opened, then the domain would be all numbers greater than or equal to -365 (or $d \geq \text-365$).

The range of a function is the set of all possible output values. Once we know the domain of a function, we can determine the range that makes sense in the situation.

• The output of function $A$ is the area of a square in square centimeters, which cannot be negative but can be 0 or greater, not limited to whole numbers. The range of $A$ is 0 and all positive numbers.
• The output of $q$ is the number of buses, which can only be 0 or positive whole numbers. If there are 120 people at the school, however, and if each bus could seat 30 people, then only up to 4 buses are needed. The range that makes sense in this situation would be any whole number that is at least 0 and at most 4.
• The output of function $v$ is the number of visitors, which cannot be fractional or negative. The range of $v$, therefore, includes 0 and all positive whole numbers.