You and your 3 friends 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?

### Finding Domain and Range

#### 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*}*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 non-negative 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* = 3*x* 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.

### Examples

#### Example 1

Earlier, you were given a problem about you and 3 friends, who are going to see a movie Friday.

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 per-ticket 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*.

#### Example 2

State the domain of each function.

- \begin{align*}y=x^{2}\end{align*}

The domain of this function is the set of all real numbers. There are no restrictions.

- \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.

- (2, 4), (3, 9), (5, 11)

The domain of this function is the set of *x* values: {2, 3, 5}.

#### Example 3

State 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 non-zero 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 4

Determine the domain of the function \begin{align*} y=\sqrt{x}\end{align*}, shown in the graph below.

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.

#### Example 5

Give an example of a function for which the domain and range are equivalent to each other.

Answers will vary. \begin{align*}y = x\end{align*} is an example.

### Review

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 20-25 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*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 1.2.