# What is the domain and range of sqrt((5x+6)/2)?

Mar 15, 2016

Domain $x \in \left[- \frac{6}{5} , \infty\right)$
Range $\left[0 , \infty\right)$

#### Explanation:

You must keep in mind that for the domain:

$\sqrt{y} \to y \ge 0$

$\ln \left(y\right) \to y > 0$

$\frac{1}{y} \to y \ne 0$

After that, you will be lead to an unequality giving you the domain.

This function is a combination of linear and square functions. Linear has domain $\mathbb{R}$. The square function though must have a positive number inside the square. Therefore:

$\frac{5 x + 6}{2} \ge 0$

Since 2 is positive:

$5 x + 6 \ge 0$

$5 x \ge - 6$

Since 5 is positive:

$x \ge - \frac{6}{5}$

The domain of the functions is:

$x \in \left[- \frac{6}{5} , \infty\right)$

The range of the root function (outer function) is $\left[0 , \infty\right)$ (infinite part can be proven through the limit as $x \to \infty$).