# Functions

## Domain and Range

Horror and thriller movies are both popular and, very often, extremely profitable. When big-budget actors, shooting locations, and special effects are included, however, studios count on even more viewership to be successful. Consider five major thriller/horror entries from the early 2000s—I am Legend, Hannibal, The Ring, The Grudge, and The Conjuring.  shows the amount, in dollars, each of those movies grossed when they were released as well as the ticket sales for horror movies in general by year. Notice that we can use the data to create a function of the amount each movie earned or the total ticket sales for all horror movies by year. In creating various functions using the data, we can identify different independent and dependent variables, and we can analyze the data and the functions to determine the domain and range. In this section, we will investigate methods for determining the domain and range of functions such as these.

### Finding the Domain of a Function Defined by an Equation

In Functions and Function Notation, we were introduced to the concepts of domain and range. In this section, we will practice determining domains and ranges for specific functions. Keep in mind that, in determining domains and ranges, we need to consider what is physically possible or meaningful in real-world examples, such as tickets sales and year in the horror movie example above. We also need to consider what is mathematically permitted. For example, we cannot include any input value that leads us to take an even root of a negative number if the domain and range consist of real numbers. Or in a function expressed as a formula, we cannot include any input value in the domain that would lead us to divide by 0.
We can visualize the domain as a “holding area” that contains “raw materials” for a “function machine” and the range as another “holding area” for the machine’s products. See .
We can write the domain and range in interval notation, which uses values within brackets to describe a set of numbers. In interval notation, we use a square bracket [ when the set includes the endpoint and a parenthesis ( to indicate that the endpoint is either not included or the interval is unbounded. For example, if a person has $100 to spend, they would need to express the interval that is more than 0 and less than or equal to 100 and write 
 We will discuss interval notation in greater detail later.
Let’s turn our attention to finding the domain of a function whose equation is provided.  Oftentimes, finding the domain of such functions involves remembering three different forms.  First, if the function has no denominator or an odd root, consider whether the domain could be all real numbers.  Second, if there is a denominator in the function’s equation, exclude values in the domain that force the denominator to be zero.  Third, if there is an even root, consider excluding values that would make the radicand negative.
Before we begin, let us review the conventions of interval notation:
1. The smallest term from the interval is written first.
2. The largest term in the interval is written second, following a comma.
3. Parentheses, ( or ), are used to signify that an endpoint is not included, called exclusive.
4. Brackets, [ or ], are used to indicate that an endpoint is included, called inclusive.
See  for a summary of interval notation.

### Using Notations to Specify Domain and Range

In the previous examples, we used inequalities and lists to describe the domain of functions. We can also use inequalities, or other statements that might define sets of values or data, to describe the behavior of the variable in set-builder notation. For example, 
 describes the behavior of 
 in set-builder notation. The braces 
 are read as “the set of,” and the vertical bar | is read as “such that,” so we would read 
 as “the set of x-values such that 10 is less than or equal to 
 and 
 is less than 30.”
 compares inequality notation,  set-builder notation, and interval notation.
To combine two intervals using inequality notation or set-builder notation, we use the word “or.” As we saw in earlier examples, we use the union symbol, 
 to combine two unconnected intervals. For example, the union of the sets 

and 

is the set 
 It is the set of all elements that belong to one or the other (or both) of the original two sets. For sets with a finite number of elements like these, the elements do not have to be listed in ascending order of numerical value. If the original two sets have some elements in common, those elements should be listed only once in the union set. For sets of real numbers on intervals, another example of a union is

### Finding Domain and Range from Graphs

Another way to identify the domain and range of functions is by using graphs. Because the domain refers to the set of possible input values, the domain of a graph consists of all the input values shown on the x-axis. The range is the set of possible output values, which are shown on the y-axis. Keep in mind that if the graph continues beyond the portion of the graph we can see, the domain and range may be greater than the visible values. See .
We can observe that the graph extends horizontally from 
 to the right without bound, so the domain is 
 The vertical extent of the graph is all range values 
  and below, so the range is 
 Note that the domain and range are always written from smaller to larger values, or from left to right for domain, and from the bottom of the graph to the top of the graph for range.

### Finding Domains and Ranges of the Toolkit Functions

We will now return to our set of toolkit functions to determine the domain and range of each.

### Graphing Piecewise-Defined Functions

Sometimes, we come across a function that requires more than one formula in order to obtain the given output. For example, in the toolkit functions, we introduced the absolute value function 
 With a domain of all real numbers and a range of values greater than or equal to 0, absolute value can be defined as the magnitude, or modulus, of a real number value regardless of sign. It is the distance from 0 on the number line. All of these definitions require the output to be greater than or equal to 0.
If we input 0, or a positive value, the output is the same as the input.
If we input a negative value, the output is the opposite of the input.
Because this requires two different processes or pieces, the absolute value function is an example of a piecewise function. A piecewise function is a function in which more than one formula is used to define the output over different pieces of the domain.
We use piecewise functions to describe situations in which a rule or relationship changes as the input value crosses certain “boundaries.” For example, we often encounter situations in business for which the cost per piece of a certain item is discounted once the number ordered exceeds a certain value. Tax brackets are another real-world example of piecewise functions. For example, consider a simple tax system in which incomes up to $10,000 are taxed at 10%, and any additional income is taxed at 20%. The tax on a total income 
 would be 
 if 
  and 
  if 
  

### Key Concepts

1. The domain of a function includes all real input values that would not cause us to attempt an undefined mathematical operation, such as dividing by zero or taking the square root of a negative number.
2. The domain of a function can be determined by listing the input values of a set of ordered pairs. See .
3. The domain of a function can also be determined by identifying the input values of a function written as an equation. See , , and .
4. Interval values represented on a number line can be described using inequality notation, set-builder notation, and interval notation. See .
5. For many functions, the domain and range can be determined from a graph. See  and .
6. An understanding of toolkit functions can be used to find the domain and range of related functions. See , , and .
7. A piecewise function is described by more than one formula. See  and .
8. A piecewise function can be graphed using each algebraic formula on its assigned subdomain. See .

### Section Exercises


### Verbal


### Algebraic

For the following exercises, find the domain of each function using interval notation.

### Graphical

For the following exercises, write the domain and range of each function using interval notation.
For the following exercises, sketch a graph of the piecewise function. Write the domain in interval notation.

### Numeric

For the following exercises, given each function 
 evaluate 
 and 


For the following exercises, given each function 
 evaluate 
 and 


For the following exercises, write the domain for the piecewise function in interval notation.

### Technology


###  Extension


###  Real-World Applications
