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# Functions

Given two sets  and , a set with elements that are ordered pairs , where  is an element of  and  is an element of , is a relation from  to . A relation from  to  defines a relationship between those two sets. A function is a special type of relation in which each element of the first set is related to exactly one element of the second set. The element of the first set is called the input; the element of the second set is called the output. Functions are used all the time in mathematics to describe relationships between two sets. For any function, when we know the input, the output is determined, so we say that the output is a function of the input. For example, the area of a square is determined by its side length, so we say that the area (the output) is a function of its side length (the input). The velocity of a ball thrown in the air can be described as a function of the amount of time the ball is in the air. The cost of mailing a package is a function of the weight of the package. Since functions have so many uses, it is important to have precise definitions and terminology to study them.

### Definition

A function  consists of a set of inputs, a set of outputs, and a rule for assigning each input to exactly one output. The set of inputs is called the domain of the function. The set of outputs is called the range of the function.

For example, consider the function , where the domain is the set of all real numbers and the rule is to square the input. Then, the input  is assigned to the output . Since every nonnegative real number has a real-value square root, every nonnegative number is an element of the range of this function. Since there is no real number with a square that is negative, the negative real numbers are not elements of the range. We conclude that the range is the set of nonnegative real numbers.

For a general function  with domain , we often use  to denote the input and  to denote the output associated with . When doing so, we refer to  as the independent variable and  as the dependent variable, because it depends on . Using function notation, we write , and we read this equation as “ equals  of .” For the squaring function described earlier, we write .

The concept of a function can be visualized:

We can also visualize a function by plotting points  in the coordinate plane where . The graph of a function is the set of all these points. For example, consider the function , where the domain is the set  and the rule is . In (Figure), we plot a graph of this function.

Every function has a domain. However, sometimes a function is described by an equation, as in , with no specific domain given. In this case, the domain is taken to be the set of all real numbers  for which  is a real number. For example, since any real number can be squared, if no other domain is specified, we consider the domain of  to be the set of all real numbers. On the other hand, the square root function  only gives a real output if  is nonnegative. Therefore, the domain of the function  is the set of nonnegative real numbers, sometimes called the natural domain.

For the functions  and , the domains are sets with an infinite number of elements. Clearly we cannot list all these elements. When describing a set with an infinite number of elements, it is often helpful to use set-builder or interval notation. When using set-builder notation to describe a subset of all real numbers, denoted , we write.

We read this as the set of real numbers  such that  has some property. For example, if we were interested in the set of real numbers that are greater than one but less than five, we could denote this set using set-builder notation by writing.

A set such as this, which contains all numbers greater than  and less than , can also be denoted using the interval notation . Therefore,.

The numbers 1 and 5 are called the endpoints of this set. If we want to consider the set that includes the endpoints, we would denote this set by writing.

We can use similar notation if we want to include one of the endpoints, but not the other. To denote the set of nonnegative real numbers, we would use the set-builder notation.

The smallest number in this set is zero, but this set does not have a largest number. Using interval notation, we would use the symbol , which refers to positive infinity, and we would write the set as.

It is important to note that  is not a real number. It is used symbolically here to indicate that this set includes all real numbers greater than or equal to zero. Similarly, if we wanted to describe the set of all nonpositive numbers, we could write.

Here, the notation  refers to negative infinity, and it indicates that we are including all numbers less than or equal to zero, no matter how small. The set

refers to the set of all real numbers.

Some functions are defined using different equations for different parts of their domain. These types of functions are known as piecewise-defined functions. For example, suppose we want to define a function  with a domain that is the set of all real numbers such that  for  and  for . We denote this function by writing.

When evaluating this function for an input , the equation to use depends on whether  or . For example, since , we use the fact that  for  and see that . On the other hand, for , we use the fact that  for  and see that .

### Finding Domain and Range

For each of the following functions, determine the i. domain and ii. range.

#### Solution

1. Consider .
1. Since  is a real number for any real number , the domain of  is the interval .
2. Since , we know . Therefore, the range must be a subset of . To show that every element in this set is in the range, we need to show that for a given  in that set, there is a real number  such that . Solving this equation for , we see that we need  such that.This equation is satisfied as long as there exists a real number  such that.Since , the square root is well-defined. We conclude that for , and therefore the range is .
2. Consider .
1. To find the domain of , we need the expression . Solving this inequality, we conclude that the domain is .
2. To find the range of , we note that since . Therefore, the range of  must be a subset of the set . To show that every element in this set is in the range of , we need to show that for all  in this set, there exists a real number  in the domain such that . Let . Then,  if and only if.Solving this equation for , we see that  must solve the equation.Since , such an  could exist. Squaring both sides of this equation, we have .
Therefore, we need,which implies.We just need to verify that  is in the domain of . Since the domain of  consists of all real numbers greater than or equal to , and,there does exist an  in the domain of . We conclude that the range of  is .
3. Consider .
1. Since  is defined when the denominator is nonzero, the domain is .
2. To find the range of , we need to find the values of  such that there exists a real number  in the domain with the property that.Solving this equation for , we find that.Therefore, as long as , there exists a real number  in the domain such that . Thus, the range is .

# Representing Functions

Typically, a function is represented using one or more of the following tools:

• A table
• A graph
• A formula

We can identify a function in each form, but we can also use them together. For instance, we can plot on a graph the values from a table or create a table from a formula.

## Tables

Functions described using a table of values arise frequently in real-world applications. Consider the following simple example. We can describe temperature on a given day as a function of time of day. Suppose we record the temperature every hour for a 24-hour period starting at midnight. We let our input variable  be the time after midnight, measured in hours, and the output variable  be the temperature  hours after midnight, measured in degrees Fahrenheit. We record our data in (Figure).

We can see from the table that temperature is a function of time, and the temperature decreases, then increases, and then decreases again. However, we cannot get a clear picture of the behavior of the function without graphing it.

## Graphs

Given a function  described by a table, we can provide a visual picture of the function in the form of a graph. Graphing the temperatures listed in (Figure) can give us a better idea of their fluctuation throughout the day. (Figure) shows the plot of the temperature function.

From the points plotted on the graph in Figure 5, we can visualize the general shape of the graph. It is often useful to connect the dots in the graph, which represent the data from the table. In this example, although we cannot make any definitive conclusion regarding what the temperature was at any time for which the temperature was not recorded, given the number of data points collected and the pattern in these points, it is reasonable to suspect that the temperatures at other times followed a similar pattern, as we can see in Figure 6.

## Algebraic Formulas

Sometimes we are not given the values of a function in table form, rather we are given the values in an explicit formula. Formulas arise in many applications. For example, the area of a circle of radius  is given by the formula . When an object is thrown upward from the ground with an initial velocity  ft/s, its height above the ground from the time it is thrown until it hits the ground is given by the formula . When  dollars are invested in an account at an annual interest rate  compounded continuously, the amount of money after  years is given by the formula . Algebraic formulas are important tools to calculate function values. Often we also represent these functions visually in graph form.

Given an algebraic formula for a function , the graph of  is the set of points , where  is in the domain of  and  is in the range. To graph a function given by a formula, it is helpful to begin by using the formula to create a table of inputs and outputs. If the domain of  consists of an infinite number of values, we cannot list all of them, but because listing some of the inputs and outputs can be very useful, it is often a good way to begin.

When creating a table of inputs and outputs, we typically check to determine whether zero is an output. Those values of  where  are called the zeros of a function. For example, the zeros of  are . The zeros determine where the graph of  intersects the -axis, which gives us more information about the shape of the graph of the function. The graph of a function may never intersect the -axis, or it may intersect multiple (or even infinitely many) times.

Another point of interest is the -intercept, if it exists. The -intercept is given by .

Since a function has exactly one output for each input, the graph of a function can have, at most, one -intercept. If  is in the domain of a function , then  has exactly one -intercept. If  is not in the domain of , then  has no -intercept. Similarly, for any real number , if  is in the domain of , there is exactly one output , and the line  intersects the graph of  exactly once. On the other hand, if  is not in the domain of  is not defined and the line  does not intersect the graph of . This property is summarized in the vertical line test.

### Rule: Vertical Line Test

Given a function , every vertical line that may be drawn intersects the graph of  no more than once. If any vertical line intersects a set of points more than once, the set of points does not represent a function.

We can use this test to determine whether a set of plotted points represents the graph of a function ((Figure)).

### Finding Zeros and -Intercepts of a Function

Consider the function .

1. Find all zeros of .
2. Find the -intercept (if any).
3. Sketch a graph of .

#### Solution

1. To find the zeros, solve . We discover that  has one zero at .
2. The -intercept is given by .
3. Given that  is a linear function of the form  that passes through the points  and , we can sketch the graph of  ((Figure)).Figure 8. The function  is a line with -intercept  and -intercept .

### Finding Zeros and -Intercepts of a Function

Consider the function .

1. Find all zeros of .
2. Find the -intercept (if any).
3. Sketch a graph of .

#### Solution

1. To find the zeros, solve . We discover that  has one zero at .
2. The -intercept is given by .
3. Given that  is a linear function of the form  that passes through the points  and , we can sketch the graph of  ((Figure)).Figure 8. The function  is a line with -intercept  and -intercept .

### Using Zeros and -Intercepts to Sketch a Graph

Consider the function .

1. Find all zeros of .
2. Find the -intercept (if any).
3. Sketch a graph of .

#### Solution

1. To find the zeros, solve . This equation implies . Since  for all , this equation has no solutions, and therefore  has no zeros.
2. The -intercept is given by .
3. To graph this function, we make a table of values. Since we need , we need to choose values of . We choose values that make the square-root function easy to evaluate.
-3-21123

Making use of the table and knowing that, since the function is a square root, the graph of  should be similar to the graph of , we sketch the graph ((Figure)).

Find the zeros of .

#### Hint

Factor the polynomial.

### Finding the Height of a Free-Falling Object

If a ball is dropped from a height of 100 ft, its height  at time  is given by the function , where  is measured in feet and  is measured in seconds. The domain is restricted to the interval , where  is the time when the ball is dropped and  is the time when the ball hits the ground.

1. Create a table showing the height  when  and . Using the data from the table, determine the domain for this function. That is, find the time  when the ball hits the ground.
2. Sketch a graph of .

#### Solution

1. Height  as a Function of Time 00.511.522.5100968464360Since the ball hits the ground when , the domain of this function is the interval .

Note that for this function and the function  graphed in (Figure), the values of  are getting smaller as  is getting larger. A function with this property is said to be decreasing. On the other hand, for the function  graphed in (Figure), the values of  are getting larger as the values of  are getting larger. A function with this property is said to be increasing. It is important to note, however, that a function can be increasing on some interval or intervals and decreasing over a different interval or intervals. For example, using our temperature function in (Figure), we can see that the function is decreasing on the interval , increasing on the interval , and then decreasing on the interval . We make the idea of a function increasing or decreasing over a particular interval more precise in the next definition.

### Definition

We say that a function  is increasing on the interval  if for all ,.

We say  is strictly increasing on the interval  if for all ,.

We say that a function  is decreasing on the interval  if for all ,.

We say that a function  is strictly decreasing on the interval  if for all ,.

For example, the function  is increasing on the interval  because  whenever . On the other hand, the function  is decreasing on the interval  because  ((Figure)).

# Combining Functions

Now that we have reviewed the basic characteristics of functions, we can see what happens to these properties when we combine functions in different ways, using basic mathematical operations to create new functions. For example, if the cost for a company to manufacture  items is described by the function  and the revenue created by the sale of  items is described by the function , then the profit on the manufacture and sale of  items is defined as . Using the difference between two functions, we created a new function.

Alternatively, we can create a new function by composing two functions. For example, given the functions  and , the composite function  is defined such that.

The composite function  is defined such that.

Note that these two new functions are different from each other.

## Combining Functions with Mathematical Operators

To combine functions using mathematical operators, we simply write the functions with the operator and simplify. Given two functions  and  we can define four new functions:

### Combining Functions Using Mathematical Operations

Given the functions  and , find each of the following functions and state its domain.

#### Solution

1. . The domain of this function is the interval .
2. . The domain of this function is the interval .
3. . The domain of this function is the interval .
4. . The domain of this function is .

For  and , find  and state its domain.

#### Solution

. The domain is .

#### Hint

The new function  is a quotient of two functions. For what values of  is the denominator zero?

## Function Composition

When we compose functions, we take a function of a function. For example, suppose the temperature  on a given day is described as a function of time  (measured in hours after midnight) as in (Figure). Suppose the cost , to heat or cool a building for 1 hour, can be described as a function of the temperature . Combining these two functions, we can describe the cost of heating or cooling a building as a function of time by evaluating . We have defined a new function, denoted , which is defined such that  for all  in the domain of . This new function is called a composite function. We note that since cost is a function of temperature and temperature is a function of time, it makes sense to define this new function . It does not make sense to consider , because temperature is not a function of cost.

### Definition

Consider the function  with domain  and range , and the function  with domain  and range . If  is a subset of , then the composite function  is the function with domain  such that.

A composite function  can be viewed in two steps. First, the function  maps each input  in the domain of  to its output  in the range of . Second, since the range of  is a subset of the domain of , the output  is an element in the domain of , and therefore it is mapped to an output  in the range of . In (Figure), we see a visual image of a composite function.

### Compositions of Functions Defined by Formulas

Consider the functions  and .

1. Find  and state its domain and range.
2. Evaluate  and .
3. Find  and state its domain and range.
4. Evaluate  and .

#### Solution

1. We can find the formula for  in two different ways. We could write.Alternatively, we could write.Since  for all real numbers , the domain of  is the set of all real numbers. Since , the range is, at most, the interval . To show that the range is this entire interval, we let  and solve this equation for  to show that for all  in the interval , there exists a real number  such that . Solving this equation for , we see that , which implies that.If  is in the interval , the expression under the radical is nonnegative, and therefore there exists a real number  such that . We conclude that the range of  is the interval .

2. We can find a formula for  in two ways. First, we could write.Alternatively, we could write.The domain of  is the set of all real numbers  such that . To find the range of , we need to find all values  for which there exists a real number  such that.Solving this equation for , we see that we need  to satisfy,which simplifies to.Finally, we obtain.Since  is a real number if and only if  is the set .

In (Figure), we can see that . This tells us, in general terms, that the order in which we compose functions matters.

Let  Let  Find .

.

### Composition of Functions Defined by Tables

Consider the functions  and  described by (Figure) and (Figure).

1. Evaluate  and .
2. State the domain and range of .
3. Evaluate  and .
4. State the domain and range of .

#### Solution

1. The domain of  is the set . Since the range of  is the set , the range of  is the set .

2. The domain of  is the set . Since the range of  is the set , the range of  is the set .

### Application Involving a Composite Function

A store is advertising a sale of  off all merchandise. Caroline has a coupon that entitles her to an additional  off any item, including sale merchandise. If Caroline decides to purchase an item with an original price of  dollars, how much will she end up paying if she applies her coupon to the sale price? Solve this problem by using a composite function.

#### Solution

Since the sale price is  off the original price, if an item is  dollars, its sale price is given by . Since the coupon entitles an individual to  off the price of any item, if an item is  dollars, the price, after applying the coupon, is given by . Therefore, if the price is originally  dollars, its sale price will be  and then its final price after the coupon will be .

If items are on sale for  off their original price, and a customer has a coupon for an additional  off, what will be the final price for an item that is originally  dollars, after applying the coupon to the sale price?

#### Hint

The sale price of an item with an original price of  dollars is . The coupon price for an item that is  dollars is .

# Symmetry of Functions

The graphs of certain functions have symmetry properties that help us understand the function and the shape of its graph. For example, consider the function  shown in (Figure)(a). If we take the part of the curve that lies to the right of the -axis and flip it over the -axis, it lays exactly on top of the curve to the left of the -axis. In this case, we say the function has symmetry about the -axis. On the other hand, consider the function  shown in (Figure)(b). If we take the graph and rotate it 180° about the origin, the new graph will look exactly the same. In this case, we say the function has symmetry about the origin.

If we are given the graph of a function, it is easy to see whether the graph has one of these symmetry properties. But without a graph, how can we determine algebraically whether a function  has symmetry? Looking at (Figure) again, we see that since  is symmetric about the -axis, if the point  is on the graph, the point  is on the graph. In other words, . If a function  has this property, we say  is an even function, which has symmetry about the -axis. For example,  is even because.

In contrast, looking at (Figure) again, if a function  is symmetric about the origin, then whenever the point  is on the graph, the point  is also on the graph. In other words, . If  has this property, we say  is an odd function, which has symmetry about the origin. For example,  is odd because.

### Definition

If  for all  in the domain of , then  is an even function. An even function is symmetric about the -axis.

If  for all  in the domain of , then  is an odd function. An odd function is symmetric about the origin.

### Even and Odd Functions

Determine whether each of the following functions is even, odd, or neither.

#### Solution

To determine whether a function is even or odd, we evaluate  and compare it to  and .

1. . Therefore,  is even.
2. . Now, . Furthermore, noting that , we see that . Therefore,  is neither even nor odd.
3. . Therefore,  is odd.

Determine whether  is even, odd, or neither.

is odd.

#### Hint

Compare  with  and .

One symmetric function that arises frequently is the absolute value function, written as . The absolute value function is defined as

Some students describe this function by stating that it “makes everything positive.” By the definition of the absolute value function, we see that if , then . Therefore, it is more accurate to say that for all nonzero inputs, the output is positive, but if , the output . We conclude that the range of the absolute value function is . In (Figure), we see that the absolute value function is symmetric about the -axis and is therefore an even function.

### Working with the Absolute Value Function

Find the domain and range of the function .

#### Solution

Since the absolute value function is defined for all real numbers, the domain of this function is . Since  for all , the function . Therefore, the range is, at most, the set . To see that the range is, in fact, this whole set, we need to show that for  there exists a real number  such that.

A real number  satisfies this equation as long as.

Since , we know , and thus the right-hand side of the equation is nonnegative, so it is possible that there is a solution. Furthermore,.

Therefore, we see there are two solutions:.

The range of this function is .

For the function , find the domain and range.

#### Solution

Domain = , range = .

#### Hint

for all real numbers .

### Key Concepts

• A function is a mapping from a set of inputs to a set of outputs with exactly one output for each input.
• If no domain is stated for a function , the domain is considered to be the set of all real numbers  for which the function is defined.
• When sketching the graph of a function , each vertical line may intersect the graph, at most, once.
• A function may have any number of zeros, but it has, at most, one -intercept.
• To define the composition , the range of  must be contained in the domain of .
• Even functions are symmetric about the -axis whereas odd functions are symmetric about the origin.

# Key Equations

• Composition of two functions
• Absolute value function