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College Preparatory Mathematics: Help and Review21 chapters | 175 lessons

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Lesson Transcript

Instructor:
*Jennifer Beddoe*

The range of a function is the set of values that are produced by that function. This lesson will define and describe how to determine the range and will also give some examples.

A **function** defines a definite relationship between an input and an output. It can be compared to a machine that spits out different items depending on what was put into the machine. An example would be a fictional toy machine. The operator puts in a plastic block, and depending on the color of that block, a certain toy was created. So, if a red block was placed in the machine, it would create a car; a blue block was a plane; a yellow block made a dog. It was the same each time; never did the machine create a car from a yellow block.

In mathematics, functions are usually equations. They can be written as equations or shown as a graph.

A function's input and output are called the domain and range of the function. The **domain** relates to the plastic blocks from the example above, and the **range** would be the toys that are created.

In this lesson, we are going to focus on the range.

The **range** of a function is the set of all possible values that can be produced by that function. Take for example, the function f(*x*) = *x*^2

No matter what value we substitute for *x*, f(*x*) will always be positive because any number multiplied to itself is positive.

Therefore, we would say that the range of this function is all positive numbers.

There are three ways to determine the range of a function: from values, from graphing, and from the rules.

A function can be written as a list of values

{(5,3), (2,7), (1,-3), (-2,9)}

The range of this simple function is the second number from each pair, or the list of output values. In this case they are 3, 7 -3, 9.

In the function example from earlier, f(*x*) = *x*^2, you also used values to determine the range. Although you might not have written them down, you took specific numbers and substituted them into the function to get an idea of what the result would be.

When you graph a function, the y-values signify the range of that function. You can often determine the range by looking at a graph.

You can see from this graph that the range of the function is between -1 and 1.

There are certain mathematical rules that can help you determine the range of a function. Unfortunately, there are also some tricky situations that require extra thought to determine the range using the rules alone.

One easy-to-understand rule is that every function where *x* is raised to an even power (*x*^2, *x*^4, *x*^6, etc.) will have a range equal to all positive numbers. Anytime you multiply a number to itself a even number of times, the result will be positive. Watch out for functions such as f(*x*) = -(*x*^2). This function will be flipped because of the negative sign and will have a range equal to all negative numbers.

Other functions that usually abide by these rules are square roots and absolute values. They have a range equal to all positive numbers as well. Just look out for negative signs and other inclusions that might change the range.

The range of a function can be written out in words, but to be more mathematically correct, it should be written as an inequality or by using interval notation. Back to our example of f(*x*) = *x*^2, writing the range as an inequality, you should have:

*y* > 0, because *y* is equal to all positive numbers.

On screen, is a range expressed using interval notation:

Let's look at a practice problem. Find the range of the following function; give your answer in words, as an inequality and using interval notation:

The range of this function is between -1 and 1.

Written as an inequality:

Interval Notation = {-1, 1}

A **function** defines a definite relationship between an input and an output. Inputs are called the **domain** and outputs are the **range**. You can determine the range of a function by inputting values, looking at a graph or following certain mathematical rules.

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College Preparatory Mathematics: Help and Review21 chapters | 175 lessons

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