# Functions

## Functions and Function Notation

A jetliner changes altitude as its distance from the starting point of a flight increases. The weight of a growing child increases with time. In each case, one quantity depends on another. There is a relationship between the two quantities that we can describe, analyze, and use to make predictions. In this section, we will analyze such relationships.

### Determining Whether a Relation Represents a Function

A relation is a set of ordered pairs. The set of the first components of each ordered pair is called the domain and the set of the second components of each ordered pair is called the range. Consider the following set of ordered pairs. The first numbers in each pair are the first five natural numbers. The second number in each pair is twice that of the first.
The domain is 
 
 The range is 
 

  
Note that each value in the domain is also known as an input value, or independent variable, and is often labeled with the lowercase letter 
    Each value in the range is also known as an output value, or dependent variable, and is often labeled lowercase letter 
   
  
A function 
  is a relation that assigns a single value in the range to each value in the domain. In other words, no x-values are repeated. For our example that relates the first five natural numbers to numbers double their values, this relation is a function because each element in the domain, 
  is paired with exactly one element in the range, 
 

  
Now let’s consider the set of ordered pairs that relates the terms “even” and “odd” to the first five natural numbers. It would appear as
  
Notice that each element in the domain, 
 
 is not paired with exactly one element in the range, 
 
For example, the term “odd” corresponds to three values from the range, 
 
 and the term “even” corresponds to two values from the range, 
 
 This violates the definition of a function, so this relation is not a function.
 compares relations that are functions and not functions. 

### Using Function Notation

Once we determine that a relationship is a function, we need to display and define the functional relationships so that we can understand and use them, and sometimes also so that we can program them into computers. There are various ways of representing functions. A standard function notation is one representation that facilitates working with functions.
To represent “height is a function of age,” we start by identifying the descriptive variables 
 
 for height and 
 
 for age. The letters 
  and 
  are often used to represent functions just as we use 
  and 
  to represent numbers and 
 
 and 
  to represent sets.
 
Remember, we can use any letter to name the function; the notation 
   shows us that 
   depends on 
  The value 
   must be put into the function 
   to get a result. The parentheses indicate that age is input into the function; they do not indicate multiplication.
We can also give an algebraic expression as the input to a function. For example 
   means “first add a and b, and the result is the input for the function f.” The operations must be performed in this order to obtain the correct result.

### Representing Functions Using Tables

A common method of representing functions is in the form of a table. The table rows or columns display the corresponding input and output values. In some cases, these values represent all we know about the relationship; other times, the table provides a few select examples from a more complete relationship.
 lists the input number of each month (January = 1, February = 2, and so on) and the output value of the number of days in that month. This information represents all we know about the months and days for a given year (that is not a leap year). Note that, in this table, we define a days-in-a-month function 
   where 
   identifies months by an integer rather than by name.
 defines a function 
   Remember, this notation tells us that 
   is the name of the function that takes the input 
   and gives the output 
   
 displays the age of children in years and their corresponding heights. This table displays just some of the data available for the heights and ages of children. We can see right away that this table does not represent a function because the same input value, 5 years, has two different output values, 40 in. and 42 in.

### Finding Input and Output Values of a Function 

When we know an input value and want to determine the corresponding output value for a function, we evaluate the function. Evaluating will always produce one result because each input value of a function corresponds to exactly one output value.
When we know an output value and want to determine the input values that would produce that output value, we set the output equal to the function’s formula and solve for the input. Solving can produce more than one solution because different input values can produce the same output value.

### Evaluation of Functions in Algebraic Forms

When we have a function in formula form, it is usually a simple matter to evaluate the function. For example, the function 
     can be evaluated by squaring the input value, multiplying by 3, and then subtracting the product from 5.

### Evaluating Functions Expressed in Formulas

Some functions are defined by mathematical rules or procedures expressed in equation form. If it is possible to express the function output with a formula involving the input quantity, then we can define a function in algebraic form. For example, the equation 
     expresses a functional relationship between 
    
    and 
  We can rewrite it to decide if 
      is a function of 
  

### Evaluating a Function Given in Tabular Form

As we saw above, we can represent functions in tables. Conversely, we can use information in tables to write functions, and we can evaluate functions using the tables. For example, how well do our pets recall the fond memories we share with them? There is an urban legend that a goldfish has a memory of 3 seconds, but this is just a myth. Goldfish can remember up to 3 months, while the beta fish has a memory of up to 5 months. And while a puppy’s memory span is no longer than 30 seconds, the adult dog can remember for 5 minutes. This is meager compared to a cat, whose memory span lasts for 16 hours.
The function that relates the type of pet to the duration of its memory span is more easily visualized with the use of a table. See .http://www.kgbanswers.com/how-long-is-a-dogs-memory-span/4221590. Accessed 3/24/2014.
At times, evaluating a function in table form may be more useful than using equations.  Here let us call the function 
 
 The domain of the function is the type of pet and the range is a real number representing the number of hours the pet’s memory span lasts. We can evaluate the function 
     at the input value of “goldfish.” We would write 
  Notice that, to evaluate the function in table form, we identify the input value and the corresponding output value from the pertinent row of the table. The tabular form for function 
     seems ideally suited to this function, more so than writing it in paragraph or function form.

### Finding Function Values from a Graph 

Evaluating a function using a graph also requires finding the corresponding output value for a given input value, only in this case, we find the output value by looking at the graph. Solving a function equation using a graph requires finding all instances of the given output value on the graph and observing the corresponding input value(s).

### Determining Whether a Function is One-to-One 

Some functions have a given output value that corresponds to two or more input values. For example, in the stock chart shown in the figure at the beginning of this chapter, the stock price was $1000 on five different dates, meaning that there were five different input values that all resulted in the same output value of $1000.
However, some functions have only one input value for each output value, as well as having only one output for each input. We call these functions one-to-one functions. As an example, consider a school that uses only letter grades and decimal equivalents, as listed in .
This grading system represents a one-to-one function, because each letter input yields one particular grade point average output and each grade point average corresponds to one input letter.
To visualize this concept, let’s look again at the two simple functions sketched in (a) and (b). The function in part (a) shows a relationship that is not a one-to-one function because inputs 
    and 
    both give output 
    The function in part (b) shows a relationship that is a one-to-one function because each input is associated with a single output.

### Using the Vertical Line Test

As we have seen in some examples above, we can represent a function using a graph. Graphs display a great many input-output pairs in a small space. The visual information they provide often makes relationships easier to understand. By convention, graphs are typically constructed with the input values along the horizontal axis and the output values along the vertical axis.
The most common graphs name the input value 
    and the output value 
    and we say 
    is a function of 
    or 
    when the function is named 
    The graph of the function is the set of all points 
    in the plane that satisfies the equation 
    If the function is defined for only a few input values, then the graph of the function is only a few points, where the x-coordinate of each point is an input value and the y-coordinate of each point is the corresponding output value. For example, the black dots on the graph in  tell us that 
    and 
    However, the set of all points 
    satisfying 
    is a curve. The curve shown includes 
    and 
    because the curve passes through those points.
The vertical line test can be used to determine whether a graph represents a function. If we can draw any vertical line that intersects a graph more than once, then the graph does not define a function because a function has only one output value for each input value. See .

###  Using the Horizontal Line Test

Once we have determined that a graph defines a function, an easy way to determine if it is a one-to-one function is to use the horizontal line test. Draw horizontal lines through the graph. If any horizontal line intersects the graph more than once, then the graph does not represent a one-to-one function.

### Identifying Basic Toolkit Functions

In this text, we will be exploring functions—the shapes of their graphs, their unique characteristics, their algebraic formulas, and how to solve problems with them. When learning to read, we start with the alphabet. When learning to do arithmetic, we start with numbers. When working with functions, it is similarly helpful to have a base set of building-block elements. We call these our “toolkit functions,” which form a set of basic named functions for which we know the graph, formula, and special properties. Some of these functions are programmed to individual buttons on many calculators. For these definitions we will use 
    as the input variable and 
    as the output variable.
We will see these toolkit functions, combinations of toolkit functions, their graphs, and their transformations frequently throughout this book. It will be very helpful if we can recognize these toolkit functions and their features quickly by name, formula, graph, and basic table properties. The graphs and sample table values are included with each function shown in .

### Key Equations


### Key Concepts

1. A relation is a set of ordered pairs. A function is a specific type of relation in which each domain value, or input, leads to exactly one range value, or output. See  and .
2. Function notation is a shorthand method for relating the input to the output in the form 
     See  and .
3. In tabular form, a function can be represented by rows or columns that relate to input and output values. See .
4. To evaluate a function, we determine an output value for a corresponding input value. Algebraic forms of a function can be evaluated by replacing the input variable with a given value. See  and .
5. To solve for a specific function value, we determine the input values that yield the specific output value. See .
6. An algebraic form of a function can be written from an equation. See  and .
7. Input and output values of a function can be identified from a table. See .
8. Relating input values to output values on a graph is another way to evaluate a function. See .
9. A function is one-to-one if each output value corresponds to only one input value. See .
10. A graph represents a function if any vertical line drawn on the graph intersects the graph at no more than one point. See .
11. The graph of a one-to-one function passes the horizontal line test. See .

### Section Exercises


### Verbal


### Algebraic

For the following exercises, determine whether the relation represents a function.
For the following exercises, determine whether the relation represents 
   as a function of 
 
 
For the following exercises, evaluate 
   at the indicated values 
 


### Graphical

For the following exercises, use the vertical line test to determine which graphs show relations that are functions.
For the following exercises, determine if the given graph is a one-to-one function.

### Numeric

For the following exercises, determine whether the relation represents a function.
For the following exercises, determine if the relation represented in table form represents 
   as a function of 
  
 
For the following exercises, use the function 
   represented in .
For the following exercises, evaluate the function 
   at the values 
   and 
  

For the following exercises, evaluate the expressions, given functions 
   and 
  
 
1.  
 

2.  
 

3.  
 


### Technology

For the following exercises, graph 
  on the given domain. Determine the corresponding range. Show each graph.
For the following exercises, graph 
   on the given domain. Determine the corresponding range. Show each graph.
For the following exercises, graph 
   on the given domain. Determine the corresponding range. Show each graph.
For the following exercises, graph 
   on the given domain. Determine the corresponding range. Show each graph.

###  Real-World Applications
