What is a Radical Function? - Definition, Equations & Graphs

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  • 0:03 What are Radical Functions?
  • 0:45 Graphing Radical Functions
  • 3:41 Quadratics
  • 5:59 Lesson Summary
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Lesson Transcript
Instructor: Cathryn Jackson

Cat has taught a variety of subjects, including communications, mathematics, and technology. Cat has a master's degree in education and is currently working on her Ph.D.

Radical functions operate very differently than regular functions. This lesson covers the definitions, equations, and graphs that you will need to know to be successful with radical functions.

What Are Radical Functions?

Elliott is a landscape architect, and he loves to design gardens with curvy shapes. He designs gardens that curve a little, a lot, or have multiple curves. Elliott has to use mathematics to make his gardens look perfect.

Elliott will have to use radical functions to graph the type of garden he wants to create. A radical function is a function that contains a square root. Radical functions are one of the few types of functions that require you to consider the domain of the function before you graph the function. The domain is the x values of a given function or relation.

Graphing Radical Functions

Elliott has a new client that wants a flower garden sweeping out from his house. Elliott knows that the function for this garden is f(x) = sqrt(4 - x). Elliott will need to graph this function so he will know the exact curve of the garden for this new client.

When you first graph a radical function, you must consider the domain of the function. Take a look at this function:


Since the domain is all of the x values of a function, we have to consider what numbers will replace this x. I would encourage you to try to work this equation along with the video. Feel free to pause it at any time to work through the problem and then play it to check your answers. If you feel like this video is too advanced, check out our other lessons for a review!

We cannot have a negative value under the square root because that would give us a complex number. You don't want to deal with complex numbers when graphing a radical function. So you need to make sure that the result of the equation underneath the square root sign is greater than zero. It is best if you work out an inequality like this: (4 - x) > 0.

The (4 - x) comes from the original equation. Everything under the square root sign needs to be considered. The result of (4 - x) needs to be greater than or equal to zero because anything less than that is a negative number. We can find the domain of the function by solving this inequality:


This means that to prevent a negative number from appearing under the square root sign, we have to pick x values that are less than or equal to 4. When graphing radical functions, it is important to graph points that are far apart from one another. This chart shows an example of x values and the resulting y value of the function:

x y
4 0
0 2
-5 3
-8 3.46
-12 4
-16 4.47

Notice that we chose x values that were less than four and far apart from one another. The graph of this radical function would look like this:


Notice that the line is curved. That's because if you graphed enough points and connected them, you would notice that radical functions always create curved lines. So remember that when you are graphing your points.

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