Rational Radical Functions

##What are rational radical functions?

Rational radical functions are essentially rational and radical functions combined.

We can think of them as rational functions, which are the quotient of two polynomials, but at least one of the polynomials is a radical function.

Let's look at an example:

$$y = \frac{x^2+2x}{\sqrt{x-4}}$$

We can see that the denominator has a radical function and is part of a rational expression.

##What is the domain and range of rational radical functions?

You can solve for the domain just like you would solve for the domain of rational and radical functions. Essentially, you want to find the values for when the function is undefined.

In this case, we know the value under the square root cannot be less than $0$, so we can set $x-4$ equal to zero and solve. We also know that for a rational function, the denominator cannot equal zero. Therefore, we know the domain in interval notation is $(4, \infty)$.

As for the range, remember that the largest power in the denominator is $\frac{1}{2}$, so there is no horizontal asymptote.

##How can we solve rational radical functions?

Well, we can solve them just like normal rational expressions.

Let's change our earlier example a bit:

$$x = \frac{x^2+2x}{\sqrt{x-4}}$$

If we rearrange the function a bit...

$$\sqrt{x-4} = \frac{x^2+2x}{x}$$

And simplify...

$$\sqrt{x-4} = x + 2$$ $${\sqrt{x-4}}^2 = (x + 2)^2$$ $$x-4 = x^2 + 4x + 4$$

Combine like terms...

$$0 = x^2 +3x +8$$

Using the quadratic equation, we find that $x = \frac{-3 - \sqrt{-23}}{2}$ and $x = \frac{-3 + \sqrt{-23}}{2}$.