# Functions

## Absolute Value Functions

Until the 1920s, the so-called spiral nebulae were believed to be clouds of dust and gas in our own galaxy, some tens of thousands of light years away. Then, astronomer Edwin Hubble proved that these objects are galaxies in their own right, at distances of millions of light years. Today, astronomers can detect galaxies that are billions of light years away. Distances in the universe can be measured in all directions. As such, it is useful to consider distance as an absolute value function. In this section, we will investigate absolute value functions.
 

### Understanding Absolute Value 

Recall that in its basic form 
    the absolute value function, is one of our toolkit functions. The absolute value function is commonly thought of as providing the distance the number is from zero on a number line. Algebraically, for whatever the input value is, the output is the value without regard to sign.

### Graphing an Absolute Value Function

The most significant feature of the absolute value graph is the corner point at which the graph changes direction. This point is shown at the origin in .
 shows the graph of 
  The graph of 
  has been shifted right 3 units, vertically stretched by a factor of 2, and shifted up 4 units. This means that the corner point is located at 
    for this transformed function.

### Solving an Absolute Value Equation

Now that we can graph an absolute value function, we will learn how to solve an absolute value equation. To solve an equation such as 
    we notice that the absolute value will be equal to 8 if the quantity inside the absolute value is 8 or -8. This leads to two different equations we can solve independently.
Knowing how to solve problems involving absolute value functions is useful. For example, we may need to identify numbers or points on a line that are at a specified distance from a given reference point.
An absolute value equation is an equation in which the unknown variable appears in absolute value bars. For example,

### Solving an Absolute Value Inequality

Absolute value equations may not always involve equalities. Instead, we may need to solve an equation within a range of values. We would use an absolute value inequality to solve such an equation. An absolute value inequality is an equation of the form
where an expression 
    (and possibly but not usually 
  ) depends on a variable 
  Solving the inequality means finding the set of all 
     that satisfy the inequality. Usually this set will be an interval or the union of two intervals.
There are two basic approaches to solving absolute value inequalities: graphical and algebraic. The advantage of the graphical approach is we can read the solution by interpreting the graphs of two functions. The advantage of the algebraic approach is it yields solutions that may be difficult to read from the graph.
For example, we know that all numbers within 200 units of 0 may be expressed as
Suppose we want to know all possible returns on an investment if we could earn some amount of money within $200 of $600. We can solve algebraically for the set of values 
 
such that the distance between 
    and 600 is less than 200. We represent the distance between 
   
   and 600 as 
    
  
This means our returns would be between $400 and $800.
Sometimes an absolute value inequality problem will be presented to us in terms of a shifted and/or stretched or compressed absolute value function, where we must determine for which values of the input the function’s output will be negative or positive.

### Key Concepts

1. The absolute value function is commonly used to measure distances between points. See .
2. Applied problems, such as ranges of possible values, can also be solved using the absolute value function. See .
3. The graph of the absolute value function resembles a letter V. It has a corner point at which the graph changes direction. See .
4. In an absolute value equation, an unknown variable is the input of an absolute value function.
5. If the absolute value of an expression is set equal to a positive number, expect two solutions for the unknown variable. See .
6. An absolute value equation may have one solution, two solutions, or no solutions. See .
7. An absolute value inequality is similar to an absolute value equation but takes the form 
     It can be solved by determining the boundaries of the solution set and then testing which segments are in the set. See .
8. Absolute value inequalities can also be solved graphically. See .

### Section Exercise


### Verbal


### Algebraic

For the following exercises, solve the equations below and express the answer using set notation.
For the following exercises, find the x- and y-intercepts of the graphs of each function.
For the following exercises, solve each inequality and write the solution in interval notation.

### Graphical

For the following exercises, graph the absolute value function. Plot at least five points by hand for each graph.
For the following exercises, graph the given functions by hand.

### Technology

For the following exercises, graph each function using a graphing utility. Specify the viewing window.

### Extensions

For the following exercises, solve the inequality.

### Real-World Applications
