Welcome to an introduction to functions.
The goals of this video are to define a function and also to identify the domain and range of a function.
A function is just a rule that for every input assigns a specific output.
You can also think of a function as a machine in which each input produces a single output.
As an example, let's say you own a prepaid cell phone.
The monthly cost is a function of the number of minutes that you use.
For example, if the cost is fifteen cents per minute, your monthly cost would be the number of minutes times fifteen cents.
So if we think of this box as the function that represents this situation, the input would be multiplied by fifteen cents.
So, for example, if you used one hundred minutes one month, one hundred times fifteen cents would be fifteen dollars.
So the input would be one-hundred minutes, and the output would be fifteen dollars.
If the next month you used two-hundred minutes, then two-hundred times fifteen cents would be thirty dollars.
So an input of two-hundred minutes has an output of thirty dollars and so on.
So we can see that for every input, there's only one output.
To think of this in a more mathematical way,
we could define two variables where one would represent the number of minutes, and one would represent the monthly cost.
The input is usually X, so we'll let X equal the number of minutes.
And we'll let Y equal the monthly cost.
So if we wanted to write an equation to represent this situation, we could say that the monthly cost, Y is equal to zero point one five times X.
However, when working with functions, we usually use something called function notation
which means instead of writing Y equals zero point one five X, we can write F of X equals zero point one five X.
So F of X is equal to Y.
And this notation here is used to show the relationship between the input and the output is a function.
So relating this notation to what we were saying earlier about the input is multiplied by fifteen cents to produce the output,
we could say that F of one-hundred, meaning one-hundred is the input into the function, and the output would be fifteen dollars.
And F of two-hundred is equal to thirty dollars.
One more connection we should make here is that each of these function values represents a point on the graph of the function.
So F of one-hundred equals fifteen dollars would represent the point where X is one-hundred and Y is fifteen.
And F of two-hundred equals thirty represents the point where X is two-hundred and the Y coordinate would be thirty.
So there's a lot going on here, but other than some different notation, it is things that we've already studied before when we had equations in terms of X and Y.
And there is quite a bit of new vocabulary when we talk about functions.
The input, usually X, is called the independent variable.
And the output, usually Y, is called the dependent variable.
Let's take a look at another situation that models a function.
Let's say that you pay seventy dollars a month for digital cable television
and then your cable company offers pay-per-view movies for three dollars per movie.
Let's say we want to write a function that models the total cost of your cable bill.
Well now there's two components to your cable bill. Let's say this first component would be the fixed cost, or the monthly cost of seventy dollars.
And then this second box represents the variable cost, or the cost that varies based upon the number of movies that you watch.
And the variable cost would be the number of movies that you watch times three dollars.
So for this situation, the input would be the number of movies that you watch that month
and the output would be the total cost for that month.
So, for example, if you watched ten movies one month, you'd have to pay seventy dollars plus ten times three dollars, or thirty dollars
which would give us a total cost of one-hundred dollars for that month.
If you watched twenty movies in one month, you'd seventy dollars plus twenty times three dollars which would be sixty dollars.
Seventy dollars plus sixty dollars would equal one-hundred and thirty dollars.
To write an equation to represent this situation, we could let X equal the number of movies watched.
And Y would be the total monthly cost.
So the equation, in terms of X and Y would be
Y, the total cost, would equal seventy dollars plus the number of movies times three dollars or three times X.
But we know this relationship is a function, so we can replace Y with F of X.
So using function notation, we would write this as F of X equals seventy plus three X.
And just to illustrate the connection once again, if we watch ten movies in one month,
we would say F of ten would equal seventy plus three times ten
which would be one-hundred dollars.
So F of ten equals one-hundred.
Now let's take a look at some additional vocabulary when dealing with functions.
The set of all possible inputs is called the domain of the function.
So the domain can be viewed as the set of all possible X values.
And the set of all possible outputs is called the range.
So we can think of the range as the set of all possible Y values.
Let's see if we can determine the domain and range for this situation.
The cost of a taxi is a four dollar flag fee plus three dollars per mile.
So the four dollar flag fee is what you're charged as soon as you enter into the taxi.
So that's the flat rate and then you have to pay three dollars per mile after that.
So if you took a taxi for one mile, you'd have four dollars plus a three dollar fee or a total of seven dollars.
If you went for two miles, you'd have the four dollar fee plus two times three dollars which would be six dollars, for a total of ten dollars.
So if we let X equal the number of miles
and Y equal the total cost
we would have Y equals the four dollar fee plus three dollars times X or the number of miles.
So using function notation, we'd have F of X equals four plus three X.
So now when we talk about the domain and the range, we're talking about what are the possible values for X
and X represents the number of miles the taxi would drive.
So we could get in the taxi and then get right out, and we'd still have a four dollar fee. So it could equal zero
but X could never be negative because we can't drive negative miles.
So the domain would be any number that is greater than or equal to zero.
Now the range are all the possible Y values.
Well looking at this cost equation, if we go zero miles we still have this four dollar flag fee for calling the taxi.
So the cost of the taxi is going to be four dollars or more.
So we could say that the range would be Y greater than or equal to four.
Let's take a look at another example of determining the domain and range from a situation that represents a function.
Here it costs five dollars to enter the theme park and two dollars for each ride.
So if you let X equal the number of rides
and, again, Y equals the total cost
we would have Y equals the five dollar cost to enter the park plus two dollars times the number of rides, or two X.
So using function notation, we replace Y with F of X. So we have F of X equals five plus two X.
So to determine the domain and range for this situation, let's go ahead and make a T table.
Remember X represents the number of rides so X could be zero,
one, two, three, and so on.
Let's see what Y would be for these different X values.
Well when X is zero, we'd have five plus two times zero, that's five.
When X is one, we'd have five plus two times one, that's seven.
When X is two, this would be four so five plus four, that would be nine.
And then when X is three, we'd have two plus this would be six or eleven and so on.
So we need to think about how are we going to express the domain and the range based upon these given X values.
Notice that X could not be a decimal or a fraction because we can't go on a decimal or a fraction number of rides.
So the domain would be the set of numbers zero, one, two, three and so on.
One way to say this would be to state the domain is all non-negative integers.
Now let's take a look at the range.
The range are the numbers five, seven, nine, eleven and so on.
So if we were to list these, it would look like this.
Now in reality, there is a maximum number of rides someone could go on in one day, therefore there's some maximum cost
but we're not given enough information in this problem so we'll go ahead and assume this could continue on forever.
So the question becomes, how do we express these values in the range.
And notice the pattern is a sequence of odd numbers. We could say all odd numbers greater than or equal to five.
So we could say that the range would be all odd numbers that are greater than or equal to five.
Okay, that'll do it for this introductory video. In the next video we'll take a look at graphing functions to determine the domain and range.