Welcome to a lesson
on discrete and continuous functions.
For review, a function is a relation
between a set of inputs, and a set of permissible outputs
with the property that each input
is related to exactly one output.
Now let's talk about discrete functions.
In a discrete function, the input values
can only assume certain values in the interval.
Usually only integers or whole numbers, but not always,
we can say the input values are counted.
In the graph of a discrete function,
only the points that have meaning to the original problem
are included in the graph.
The graph of a discrete function
may look like a scatter plot, as shown here below.
Now let's talk about continuous functions.
A continuous function allows the input values
to assume any value in a given interval,
and the points on the graph are connected
with a line or a curve.
We can also say the input values are measured.
You can draw the graph of a continuous function
without ever having to lift your pencil off the paper.
Here's an example of a continuous function.
Notice how all of the points are connected.
Now let's look at some examples of continuous functions,
and discrete functions.
The table below gives three examples
of continuous and discrete functions,
along with the input and output quantities for each.
For continuous functions, number one,
the height of a ball as it is thrown through the air
where the input quantity is time in seconds,
and the output quantity is height in feet.
Number two, the speed of a truck as it drives
along the highway,
where the input quantity is time in seconds,
and the output quantity is speed in miles per hour.
And number three, the breaking distance required
by a cement truck to come to a complete stop.
The input quantity is the speed of the truck
in miles per hour,
and the output quantity is the breaking distance in feet.
For these three functions,
the inputs can take on any value over a given interval,
and all the points of these three graphs would be connected.
Now let's go to our examples of discrete functions.
Number one, the number of felony arrests
in a town each year,
where the input quantity is years since 2005,
the output quantity is the number of felony arrests.
Number two, the cost of food at a wedding.
The input quantity is the number of people
attending the wedding,
and the output quantity is the cost of the food.
And then number three, we have the weight
of a bag of oranges as more oranges are added to the bag.
The input quantity is the number of oranges,
and the output quantity is the weight of the bag of oranges.
For all three of these functions, the input values
only take on certain values over the given interval.
Notice how for the years since 2005,
the number of people attending the wedding,
and the number of oranges in the bag,
can only be whole numbers
which is why these are discrete functions.
Now let's look at some problems.
Gasoline costs $2.50 per gallon.
The total cost of the gas depends on the number
of gallons pumped.
Complete the table below.
Looking at the table in the first column,
we have gallons of gas represented by the variable g.
In the second column we have the cost in dollars,
represented by the variable c.
Because the cost of the gas is $2.50 per gallon,
to determine the cost,
we multiply $2.50 times the number of gallons pumped.
Which means if zero gallons of gas are pumped,
the total cost is zero dollars.
If two gallons of gas are pumped,
the cost is $2.50 times two, which is five dollars.
If four gallons of gas are pumped,
the cost is $2.50 times four, which is 10 dollars.
Now we could keep going here,
but there's also a pattern in the table.
Notice how each time the number of gallons increases by two,
the cost in dollars increases by five.
So the cost of six gallons is 15 dollars,
the cost of eight gallons is 20 dollars,
the cost of 10 gallons is 25 dollars,
and the cost of 12 gallons is 30 dollars.
Next, does the table represent a continuous
or a discrete function?
Looking at the input values,
notice how all of these are measurements,
which means the number of gallons can take on
any possible value over a given interval,
and therefore the inputs are continuous.
And because the points of this graph aren't connected,
this table represents a continuous function.
Next, write an equation that models the situation.
And we will be using the variable c for cost in dollars,
and g for the gallons of gas.
And because the cost is $2.50 per gallon,
our equation is c, the cost in dollars,
equals $2.50, or 2.5,
times g, the number of gallons of gas.
We should make a note that g can never be negative.
So let's say g is greater than or equal to zero.
The last part is to graph the equation.
We will do this by applying the points from the table.
Each row represents an ordered pair,
and each ordered pair is a point on the graph.
The first ordered pair is 0, 0, which is the origin.
The next point is 2, 5.
From the origin, write two up five.
The next ordered pair is 4, 10, write four up ten.
Next ordered pair is 6, 15, write six up 15, and so on.
And because we know we have a continuous function
to form the graph, we connect the points.
This is the graph of the continuous function.
Let's look at one more example.
You are put in charge of reserving hotel rooms
for you and your classmates' field trip to New York.
Each room costs $150 per night, including all taxes.
The trip will be for two nights.
You will need to reserve three to eight rooms.
Complete the table below.
You could be a little careful here
because each room costs $150 per night,
but the stay is for two nights,
which means each room will cost $300 for the two nights.
Looking at the table, the first column
is the number of rooms, represented by the variable n.
And the second column is the total cost in dollars,
represented by the variable c.
To complete the table, we first need to determine
the total cost if three rooms are reserved for two nights.
Well, because it costs $300
to reserve one room for two nights, to reserve three rooms
it's going to cost $300 times three, or $900.
To reserve four rooms,
it's going to cost $300 times four, or $1200.
To reserve five rooms for two nights,
it's going to cost $300 times five, or $1500.
Now, I could keep going here,
but notice how there is a pattern in the table.
Each time the number of rooms increased by one,
the total cost increases by $300.
Which means the cost of six rooms for two nights
is going to be $1500 plus $300, or $1800.
The cost for seven rooms for two nights
is going to be $1800 plus $300, or $2100.
And the cost for eight rooms for two nights
is going to be $2100 plus $300, or $2400.
The next question is, does the table represent
a continuous, or a discrete function?
Looking at the inputs from the table,
because the number of rooms can only assume certain values
over a given interval,
or in this case whole numbers from three to eight,
this is an example of a discrete function.
The values between these values of n don't make sense
in this situation because we can not reserve
a fraction or decimal part of a room,
making this a discrete function.
Next, we're asked to write an equation
that models the situation.
To write the equation, we will use n for the number of rooms
and c for the total cost.
And because it costs $300
to reserve one room for two nights,
the total cost c is equal to $300 times n, or 300n,
but n must be in this set of whole numbers
from three to eight.
And then finally we're asked to graph the equation.
Again, just keep in mind, we have a discrete function,
and therefore our graph will only be a set of points,
and the ordered pairs from the table
represent the points on the graph.
The first ordered pair is 3, 900.
From the origin we move right three units,
and up 900 units.
The next ordered pair is 4, 1200.
Write four, up 1200.
The next ordered pair is 5, 1500.
And so on.
6, 1800.
7, 2100.
And finally, 8, 2400.
Again, we have a discrete function,
we do not connect the points.
I hope you found this helpful.