1.2: Domain and Range
You and your 3 friends are are debating which theater to choose for a trip to the movies on Friday. There are 4 theaters in town, all with different prices for tickets. The most expensive theater charges $12.50 per person, but has stadium seating and a screen 5 stories tall. The cheapest is only $5.50 per ticket, but only plays older movies and is much less comfortable. The other two are between the extremes at $8.50 and $9.75 each.
If you were to graph the total costs for the 4 of you based on which theater you choose, which values would represent the domain and which the range?
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James Sousa Determining Domain and Range
Guidance
Independent variable, domain
The domain of a function is defined as the set of all x values for which the function is defined. For example, the domain of the function \begin{align*}y=3x\end{align*} is the set of all real numbers, often written as \begin{align*}\mathbb R\end{align*}. This means that x can be any real number. Other functions have restricted domains. For example, the domain of the function \begin{align*}y = \sqrt{x}\end{align*} is the set of all real numbers greater than or equal to zero. The domain of this function is restricted in this way because the square root of a negative number is not a real number. Therefore, the domain is restricted to nonnegative values of x so that the function values will be defined.
The variable x is often referred to as the independent variable, while the variable y is referred to as the dependent variable. We talk about x and y this way because the y values of a function depend on what the x values are. That is why we also say that “y is a function of x.” For example, the value of y in the function y = 3x depends on what x value we are considering. If x = 4, we can easily determine that y = 3(4) = 12.
Dependent Variable, Range
The range of a function is defined as the set of all y values for which a function is defined. Just as we did with domain, we can examine a function and determine its range. Again, it is often helpful to think about what restrictions there might be, and what the graph of the function looks like. Consider for example the function \begin{align*}y=x^{2}\end{align*}. The domain of this function is \begin{align*}\mathbb R\end{align*}, all real numbers, but what about the range?
The range of the function is the set of all real numbers greater than or equal to zero. This is the case because every y value is the square of an x value. If we square positive and negative numbers, the result will always be positive. If x = 0, then y = 0. We can also see the range if we look at a graph of \begin{align*}y=x^{2}\end{align*}. Notice below that the y values are all greater than or equal to zero.
Example A
State the domain of each function:
a. \begin{align*}y=x^{2}\end{align*}  b. \begin{align*}\frac{1}{x}\end{align*}  c. (2, 4), (3, 9), (5, 11) 

Solution:
a. \begin{align*}y=x^{2}\end{align*}
The domain of this function is the set of all real numbers. There are no restrictions.
b. \begin{align*}\frac{1}{x}\end{align*}
The domain of this function is the set of all real numbers except x = 0. The domain is restricted this way because a fraction with denominator zero is undefined.
c. (2, 4), (3, 9), (5, 11)
The domain of this function is the set of x values: {2, 3, 5}.
Example B
State the domain and range of the function \begin{align*}y=\frac{2}{x}\end{align*}
Solution: The domain and range of the function \begin{align*}y=\frac{2}{x}\end{align*}
For this function, we can choose any x value except x = 0, since we cannot divide a number by zero. Therefore the domain of the function is the set of all real numbers except x = 0.
The range is also restricted to the nonzero real numbers, but for a different reason. Because the numerator of the fraction is 2, the numerator can never equal zero, so the fraction can never equal zero.
Example C
Determine the domain of the function \begin{align*} y=\sqrt{x}\end{align*}, shown in the graph below.
Solution
First consider what restrictions there might be and then look at the graph. We can see that \begin{align*} y=\sqrt{x}\end{align*} has a domain of all real numbers greater than or equal to zero because the graph only exists for x values that are positive, since there are no real square roots of negative numbers.
Concept question wrapup If you were to graph the total costs for the 4 of you based on which theater you choose, which values would represent the domain and which the range? One function that describes this situation would be: \begin{align*}P(t) = 4t\end{align*}, where \begin{align*}P(t)\end{align*} would be total Price based on perticket price, and it is equal to 4 x ticket price. In this case, your domain would be: {$5.50, $8.50, $9.75, $12.50} since these are the prices you would input in place of the independent variable, x to get the total price for each theater. The range would be: {$22.00, $34.00, $38.00, $50.00} since these are the output values given by the dependent variable, y. 

Vocabulary
The domain of a function is the set of all x values under which the function is defined. The x variable is also called the independent variable.
The range of a function is the set of all y values under which the function is defined. The y variable is also called the dependent variable.
Guided Practice
Questions
 1) Determine if each relation is a function:


a) (1,4), (0, 3), (1, 5), (1, 7), (2, 15) b) \begin{align*}y=x\end{align*}

 2) State the domain and range of each relation in question A.
 3) Give an example of a function for which the domain and range are equivalent to each other.
 4) Imagine a playground full of children where all of the boys are wearing baseball caps. The function "boys in the playground" could be used to identify kids wearing hats. In this case, which group would be the domain, and which the range? In other words, what group of kids is the input, and what group the output?
 5) Would the example from question 4 be a function if used backwards? Which group would be the domain and which the range in that case?
Answers

1) a. Not a function, since there are two different y values for the same x value (1).
 b) Is a function. Since x and y are the same number (equal), any number chosen for x will have one and only one y.

2) a) The domain is the set of x values: {1, 0, 1, 2}. The range is the set of y values: {3, 4, 5, 7, 15}
 b) The domain or x value can be any real number. D:\begin{align*}\mathbb R\end{align*}. The range or y value can therefore also be any real number. R: \begin{align*}\mathbb R\end{align*}
 3) Answers will vary. \begin{align*}y = x\end{align*} is an example.
 4) In this example, "boys in the playground" is the domain, since you would use that group to identify hatwearers.
 5) It could be, it depends on whether or not there are girls wearing hats! If there are girls in the playground wearing hats, then inputing "kids wearing hats" would not return only "boys in the playground". In either case, however, "kids wearing hats" would be in domain in this version of the function.
Practice
Determine the Domain and the Range of the relations.
 Relation: {(0,4) (3, 20) (90, 33)}
 Relation: {(3, 4) (6, 37) (10, 10) (31, 2)}
 Tina’s car travels about 30 miles on one gallon of gas. She has between 10 and 12 gallons of gas in the tank. Find the domain and range of the function to calculate how far she can drive.
 Joe and three of his friends plan on going bowling and plan on bowling one or two games each. Each game costs $2.75. Find the domain and range of the function calculating the cost of the trip.
 Bob had a summer job that paid $10.00 per hour and he worked between 2025 hours every week. His weekly salary can be modeled by the equation S = 10h, where S is his weekly salary and h is the number of hours he worked per week. What is the independent variable for this problem? Describe the domain and range for this problem.
 What does each value in the ordered pair (20, 200) mean in context of the previous problem?
 Which group of students represents the domain, and which the range, in these ordered pairs? (Jim, Kitty) (Joe, Betty) (Brian, Alice) (Jesus, Anissa) (Ken, Kelli)
State the domain and range:
 \begin{align*}y = x\end{align*}
 \begin{align*}x = y\end{align*}
 \begin{align*}y = x^2\end{align*}
 \begin{align*}x = y^2\end{align*}
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Term  Definition 

closed interval  A closed interval includes the minimum and maximum values (endpoints) of the interval. 
dependent variable  The dependent variable is the output variable in an equation or function, commonly represented by or . 
domain  The domain of a function is the set of values for which the function is defined. 
Function  A function is a relation where there is only one output for every input. In other words, for every value of , there is only one value for . 
independent variable  The independent variable is the input variable in an equation or function, commonly represented by . 
open interval  An open interval does not include the endpoints of the interval. 
Range  The range of a function is the set of values for which the function is defined. 
Relation  A relation is any set of ordered pairs . A relation can have more than one output for a given input. 
union  is a symbol that stands for union and is used to connect two groups together. It is associated with the logical term OR. 
Image Attributions
Here you will learn to identify the independent variable, (the domain) and the dependent variable (the range) of a function.