March 31, 2021

Chapters

## Linear Function Definition

**exemplify**this with linear functions. A function takes a number as an input and gives a transformation of that number as an output.

**goal**of functions is to show the relationship between the input and the output. This can be seen easily if we plot a function onto a graph.

x | f(x) |

... | ... |

-1 | 1 |

0 | 4 |

1 | 7 |

... | ... |

What you can see in this **graph** is a linear function. Here is the standard way to write a linear function.

f(x) | Read as ‘f of x’ and can be thought of as y |

m | The slope |

x | The input |

b | The y-intercept |

When we use functions, we use f of x, written as **f(x),** which can be thought of as what we usually call y. The linear function we used above can be written as:

## Radicals

A radical function is any function that includes **radicals.** In algebra, radicals are also called roots. You may be familiar with how they look like:

This **symbol** is quite powerful - let’s take a look at a few examples of radicals to see why.

Name | Example | |

, | Square root, square | |

Cube root, cubic | ||

Root |

A radical function, then, is simply a function that takes any number as an input and takes a root of it. Let’s take a **basic** example:

The number inside of a **square root,** so we can plug in any number greater or equal to zero.

x | f(x) |

0 | 0.0 |

1 | 1.0 |

2 | 1.4 |

3 | 1.7 |

... | ... |

We can keep going, plotting each output up to **21**. Next, we can graph the result.

As you can see, the graph of a radical function is quite **different** from a linear function. Take a look at some more examples below.

## Power Function

A power function is a function that involves a power. A power is also known as taking a number to the **nth** power. Take a look at the image below to get a better understanding.

As you can see, we have a couple of elements involved here. The table below explains each element in the **power** function.

k | Constant | |

x | Variable | x,y,z,etc. |

n | Power |

When you’re using power functions, you will probably come across **two terms:**

**Growth**function**Decay**function

These two functions are found in many different disciplines and are usually used to model things with an **exponential relationship.** An exponential relationship is one modelled by an exponent, also known as a power. Take a look at a simple exponent relationship below.

As you can see, **‘growth’** happens super quickly. While there’s not that much of a difference between 3 numbers, the difference between 9 to the power of 3 and 10 to the power of 3 is almost 300.

## Radical and Power Rules

There are many rules when it comes to radicals and powers. First, let’s focus on **radical rules.** Take a look at the table below to get an idea of the rules and examples for each rule.

Rule | Example | |

1 | = | = |

2 | ||

3 | = |

Now that you understand a bit more about radical rules, let’s take a look at some rules for **powers.**

Rule | Example | |

1 | ||

2 | ||

3 | ||

4 | ||

5 |

## Step-by-Step Factoring Radicals

Now that you’ve learned about radicals and how they relate to powers, let’s take a look at an example where we will factor a radical **step-by-step.** First, we will start with an easier example. The three examples after this one build upon each other.

Take a look at the radical below.

While this may look complex at first, we can start by focusing on what is inside of the **fraction.** The first step in simplifying this radical is to recall that negative exponents mean we have to take the inverse of the number.

Next, since we are multiplying a whole number by a fraction, we can **rewrite** that into one simple fraction.

Now, we can see that since we have the same main number in the **numerator** and the **denominator,** we can use the power rules to simplify this fraction.

Since the nth square root of a number to the nth power is just the number, we can **stop** here and say the answer is 2.

## Example 1

In this example, try to simplify the **radical** below by yourself first. Then, compare your solution with the following.

Take a look at the **step-by-step** solution below.

## Example 2

In this example, try to simplify the radical below by yourself first. Then, **compare** your solution with the following.

Take a look at the **step-by-step** solution below.

## Example 3

In this example, try to **simplify** the radical below by yourself first. Then, compare your solution with the following.

Take a look at the step-by-step solution below.