# Algebra: What Are Radical Functions?

**Radical Functions**

If we have an understanding of functions, then we would know that function has an input, output and a relationship. We can also recall that a function is usually denoted by “*f*(x) = …”. With radical functions, you have to consider the domain carefully and we will get to know why. Radical functions are functions that has a square root sign and it is because of the root sign that we have to note that certain numbers cannot be part of the domain.

**Example 1**

*f*(*x*) = √3*x* – 6

As we examine the function, if we should evaluate the function for *f*(1) we will have:

*f*(1) = √3(1) – 6

= √3 – 6

= √-3

Since you cannot find the root of a negative number, then an answer does not exist. Therefore, now number that is less than 2 cannot be part of the domain.

**Example 2**

What is the domain for the following function?

In order to determine the domain for the function, we have to recall a critical concept

Recall √-a = Undefined

Therefore, the domain will exclude all the values that when it is plugged into the function a negative answer is produced. Therefore, from the question, once the there is a negative number in the denominator, then the function is undefined. Let us try with some random numbers.

* f *(-1)

By just working and ignoring the root sign for now:

Since a negative divided by a negative is positive then the final answer is √1/2 and since we get a positive number then -1 is part of the domain.

* f*(0)

By just working and ignoring the root sign for now:

So the final answer is √-2 which is undefined, therefore 0 is not part of the domain. 1 is also not part of the domain because of we plug in 1, we will end up with zero in the denominator and anything divided by zero is undefined. But any other value we place in the function we will end up with a positive value: therefore the domain is:

**-∞ ≤ x ≤ -1 U 1 < x ≤ ∞**

The above is interpreted as *x* can be from all the values from negative infinity to negative one and all the values greater than one to positive infinity. Notice that you see a union notation. That is because there are some values that are excluded because the domain cannot be all the values from negative infinity to positive infinity. Therefore you have to have the union notation that shows that it is a combination of the two separate domain so as to exclude 0.

**Graphing Radical Functions**

Sometimes you may be required to graph radical functions. If you understand the concept of domain and range then you should be able to graph any radical function. Let us graph the following:

In order to solve this we can set up a table and use a suitable domain if you are not given one:

If we plug in the values for *x* in the equation we should get these values:

x |
1 | 2 | 3 | 4 | 5 |

y |
0 | 1.7 | 2.8 | 3.9 | 4.9 |

Below is the graph of the function, if you notice from the function, if *x* = 0, then the function is undefined. So the domain is from 1 to positive infinity. Thus we can write it as 0 < *x*. * *