# How do you find the domain and range of 1 / root4 (x^2 -5x)?

Dec 31, 2017

Domain: $\left\{x | x < 0 , x > 5\right\}$
Range: $y \in {\mathbb{R}}^{+}$

#### Explanation:

Finding the Domain
To find the domain we want to look at where the function goes undefined. The first thing that might spring to mind is if the denominator equals zero. To find when that happens, we can solve this equation:

$\sqrt{{x}^{2} - 5 x} = 0$

${\left(\sqrt{{x}^{2} - 5 x}\right)}^{4} = {0}^{4}$

${x}^{2} - 5 x = 0$

$x \left(x - 5\right) = 0$

By the zero factor principle we get that the solutions are $x = 0$ and $x = 5$.

The other possibility that might make the function undefined is if the bit inside the 4th root is negative. To find when this occurs, we solve this inequality:

${x}^{2} - 5 x < 0$

$x \left(x - 5\right) < 0$

To find the negative possibilities for this inequality, we need to look at the intervals between the zeroes since that is where the function could potentially go negative. For the function to be negative, only one of the products may be negative.

On the interval $\left(- \infty , 0\right)$ both factors will be negative, so the product will be positive.

On the interval $\left(0 , 5\right)$ one of the factors will be negative, so we get a positive product.

And finally $\left(5 , \infty\right)$ will always make two positive factors and a positive product.

This means that the function is undefined on the interval $\left(0 , 5\right)$. Combining this with the zeroes from the denominator we get an undefined interval of $\left[0 , 5\right]$ and therefor our domain is:
$\left\{x | x < 0 , x > 5\right\}$

Finding the Range
The vertical asymptotes caused by the zeroes in the denominator would normally go to $\infty$ from the positive direction and $- \infty$ from the negative direction. However, the root causes the negative values to be undefined, so we only get values greater than $0$ to $\infty$ (note that the horizontal zero asymptotes never actually reach $0$).

This means that our will be all the positive real numbers, ${\mathbb{R}}^{+}$