In this lesson, we'll review the domain and range of a function. Then, we'll look at examples of applying the domain and range in real-world settings and explore the information that the domain and range can provide in day-to-day situations.
Domain and Range
Let's start with an example. Suppose the gas tank in Zack's car holds 20 gallons of gas, and the car gets 32 miles per gallon. Zack fills up his tank with gas and heads out on a trip. His distance (or D) traveled on one tank of gas can be represented by the function:
|D(x) = 32x, where x is the number of gallons of gas he used
The formula would be D(x) = 32x, with 0 less than or equal to x, which is less than or equal to 20.
In this function, the number of gallons used can be anywhere from 0 gallons to 20 gallons, since the tank holds 20 gallons of gas. Based on this and the fact that the car gets 32 miles per gallon, Zack can drive anywhere from 0*32 = 0 miles to 20*32 = 640 miles on one tank of gas.
Mathematically speaking, the number of gallons Zack can possibly use would be the domain of the function, and the possible number of miles traveled would be the range of the function. The domain consists of the values that can be put into a function that make sense (that is, values that don't make the function undefined). In other words, Zack can put up to 20 gallons into his tank: It wouldn't make sense to put 21 gallons in because one would spill out. The range of a function is the set of values that come out of a function based on the function's domain.
Let's take a look at our function D(x) = 32x and put this into context. As we said, the domain is the possible number of gallons of gas used. This is represented by x in the function, or the input value. Makes sense, doesn't it? The input values are the domain of the function, so the number of gallons used is the domain. Similarly, the output values, or D(x), represent the range, so the range is the possible distance traveled.
To apply the domain and range in real-world settings, we take a function that represents a real-world situation and then analyze what the domain and range represent in the function. This allows us to apply the domain and range in a real-world setting.
Consider the following scenario. You're throwing a surprise birthday party for a friend of yours. You head to the store with $15 in your pocket to buy some chips to go with a dip you made. When you get there, you see that each bag of chips costs $3. So, the number (or N) of bags of chips you buy can be represented by the function N(x) = x / 3, where x is the amount of money you can spend.
This is an interesting example, because there are a few things to consider. First, let's talk about the function's domain. In the function, N(x) = x / 3, the domain consists of all values of x that we can plug in the function. Just looking at the function , it would appear that x can be any real number. However, if we put our function into the context described, the domain is limited. We saw that you only have up to $15 to spend. Therefore, we can only plug values from 0 to 15 into the function.
Not only that, but you are buying only chips, and each bag costs $3. You can't buy a fraction of a bag of chips, so really, you can only plug in multiples of 3. Putting all this together, we have that the domain of the function representing this real-world situation consists of the numbers 0, 3, 6, 9, 12, and 15. Figuring out the domain of this function in the given context tells us how much money we have to spend and what the different total cost amounts may be.
Now, let's consider the range. Since we are only plugging in the values 0, 3, 6, 9, 12, and 15, it follows that the range will consist of all the outputs created by inputting these values into the function.
||N(x) = x / 3
We see that the possible outputs created by our domain are 0, 1, 2, 3, 4, and 5. This is our range. The range tells us how many bags of chips we can get based on how much money we have and how much money we spend.
Once again, we see that the domain and range provide extremely important information about the real-world situation of purchasing a product with a limited amount of money.
The domain of a function consists of all the values that we can plug into a function that make sense. In other words, they are the values that don't make the function undefined. The range of a function consists of all of the outputs created by plugging the domain values into the function.
When we have a function representing a real-world situation, the domain and range can be determined based on the context of the problem. This may result in restrictions on the domain and range, and the values in both the domain and range can help us to better understand the problem and give us useful information about the real-world scenario. This is why being able to analyze a real-world function's domain and range is very important and useful in mathematics. It's really easy: just remember that domain = inputs and range = outputs!