# How do you find the domain and range of f(x)=sqrt(x-4)?

Aug 6, 2017

Here's what I got.

#### Explanation:

You know that when working with real numbers, you can only take the square root of a positive number.

This implies that the domain of the function, which includes all the values that $x$ can take for which $f \left(x\right)$ is defined, will have to account for the fact that

$x - 4 \ge 0$

This is equivalent to saying that

$x \ge 4$

You can thus say that the domain of this function is all real numbers that satisfy the above condition. In interval notation, this will be $x \in \left[4 , + \infty\right)$.

The range of the function tells you the values that the function can take for values that $x$ can take.

In this case, if you take the square root of a positive number, you will end up with a positive number, so

$f \left(x\right) = \sqrt{x - 4} \ge 0 \textcolor{w h i t e}{.} \left(\forall\right) x \in \left[4 , + \infty\right)$

The minimum value that $f \left(x\right)$ can take occurs when $x = 4$, so

$f \left(4\right) = \sqrt{4 - 4} = \sqrt{0} = 0$

For any other value of $x > 4$, you will have $f \left(x\right) > 0$. In interval notation, the range of the function is $\left[0 , + \infty\right)$.

graph{sqrt(x-4) [-10, 10, -5, 5]}