Consider the polynomial . A good question to ask would be whether the function is invertible. To help us decide, here is the graph of .

Notice that this graph does not pass the Horizontal Line Test, so the function is not one-to-one, and therefore not invertible.

Our polynomial has many terms, so to simplify the situation, we’ll look only at polynomials of the form , where is a positive integer.

Odd Roots

Recall that every polynomial , where is odd, has the same basic shape. This is demonstrated in the figure below by the graphs of for .

To see more of how these graphs change with , follow the .

Now, are these functions invertible? Looking at the graphs, we see that these functions pass the Horizontal Line Test. Thus, the functions are one-to-one, and therefore invertible.

Let’s delve more deeply into the example. We have now established that it is invertible, and it’s inverse is .

Draw both functions on the axes provided, then answer the following questions about the function .
(a)
What is the -intercept of ?
(b)
What is the -intercept of ?
(c)
What is the domain of ?
(d)
What is the range of ?
(e)
As goes to , goes to .
(f)
As goes to , goes to .
(g)
Does this function have any vertical asymptotes? yesno

Recall that a function is even if for all in its domain, and is odd if for all . Otherwise, the function is neither. Let’s consider an example. Given , and , since . Based on this example, do you think is even, odd, or neither?

If you guessed odd, then you are correct! All odd-index root functions are odd functions.

Even Roots

We again begin by recalling the general shape of , but this time for even. These functions also have the same basic shape for all even . This is demonstrated by the graphs of for given below.

To see more of how these graphs change with , follow the .

Now, are these functions invertible? All of the graphs in the figure above are symmetric about the -axis (example ), so they do not pass the horizontal line test. Thus, these functions are not one-to-one, and therefore not invertible.

So, how can we define an even root function? For example, what does really mean, and how is it related to ?

Consider the polynomial , graphed below with its inverse relation .

Observe that by restricting the domain of to , we now have a function which passes the horizontal line test, and can thus be inverted. The following picture illustrates the situation.

Now, our inverse relation is actually a function, since it passes the Horizontal Line Test. Therefore, if we let for , we can then define as the inverse function of on this restricted domain.

We now repeat Exploration 1 for . Draw both functions on the axes provided, then answer the following questions about the function .
(a)
What is the -intercept of ?
(b)
What is the -intercept of ?
(c)
What is the domain of ?
(d)
What is the range of ?
(e)
As goes to , goes to .
(f)
Does this function have any vertical asymptotes? yesno

One question we might ask is whether and are truly inverses. The answer may seem like an obvious “yes!”, but since we restricted the domain of in order to define , we need to check. To check whether and are inverses, we need to confirm that and . That is, when we plug in a number to and , we should get the same number as the output. Let’s try plugging in to . This gives us which is not the same as . If we repeat this process with a few more numbers, we find that , , , and . We can conclude that is a function that takes its input and returns its absolute value. That is, . Since , we conclude that does not output its input, and therefore, and are not inverses. This is something that will be extremely important when solving equations using even roots.

Now, what if we instead restricted our domain to ? Consider defined for . The graph of this function is below.

By the Horizontal Line Test, this restriction is one-to-one, and therefore invertible. The inverse of this function as shown above is .

Using Roots to Solve Equations

If we are asked to find all values such that , then the question is asking which values of multiplied by themselves give 4. In other words, find such that is equal to 4. It is simple to see that there are two values which make this true: In solving an equation, it is common to express this as follows.

Since is not one-to-one, there are two values of which make it equal to any positive number, as demonstrated in the following graph.

These examples illustrate a general principle that is good to have in your toolbox for solving equations. If , then we know that . This is true for any odd powers. However, if , then either or . This is true for any even powers.

Finding -intercepts of a Quadratic in Vertex Form

Now, we can use our understanding of the squareroot function to find the -intercepts of a quadratic given in vertex form.

Notice that this gives us a third method for finding the roots (-intercepts) of a quadratic in general. We can use any of these methods to solve a quadratic.

(a)
Factor the quadratic and write it in Root Form
(b)
Use the quadratic formula to find the roots
(c)
Write the quadratic in vertex form and then solve using a squareroot

Mathematically, these last two methods are actually related. The quadratic formula is just what happens when you rewrite the general quadratic in vertex form and then solve for !

Composing and

This final example is going to be a very important one that comes up often so we we will give it its own section.