# What is the domain of the expression sqrt(7x+35)?

Jun 10, 2017

Domain: From $- 5$ to infinity
$\left[- 5 , \infty\right)$

#### Explanation:

The domain means the values of $x$ that make the equation untrue. So, we need to find the values that $x$ cannot equal.

For square root functions, $x$ cannot be a negative number. $\sqrt{- x}$ would give us $i \sqrt{x}$, where $i$ stands for imaginary number. We cannot represent $i$ on graphs or within our domains. So, $x$ must be larger than $0$.

Can it equal $0$ though? Well, let's change the square root to an exponential: $\sqrt{0} = {0}^{\frac{1}{2}}$. Now we have the "Zero Power Rule", which means $0$, raised to any power, equals one. Thus, $\sqrt{0} = 1$. Ad one is within our rule of "must be greater than 0"

So, $x$ can never bring the equation to take a square root of a negative number. So let's see what it would take to make the equation equal zero, and make that the edge of our domain!

To find the value of $x$ the makes the expression equal to zero, let's set the problem equal to $0$ and solve for $x$:

$0 = \sqrt{7 x + 35}$

square both sides

0^2 = cancelcolor(black)(sqrt(7x+35)^cancel(2)

$0 = 7 x + 35$

subtract $35$ on both sides

$- 35 = 7 x$

divide by $7$ on both sides

$- \frac{35}{7} = x$

$- 5 = x$

So, if $x$ equals $- 5$, our expression becomes $\sqrt{0}$. That is the limit of our domain. Any smaller numbers than $- 5$ would give us a square root of a negative number.