What is the domain and range of r(x)= -3sqrt(x-4) +3?

Aug 21, 2015

Domain: $\left[4 , + \infty\right)$
Range: $\left(- \infty , 3\right]$

Explanation:

Your function is defined for any value of $x$ that will not make the expression under the square root negative.

In other words, you need to have

$x - 4 \ge 0 \implies x \ge 4$

The domain of the function will thus be $\left[4 , + \infty\right)$.

The expression under the square root will have a minimum value at $x = 4$, which corresponds to maximum value of the function

$r = - 3 \cdot \sqrt{4 - 4} + 3$

$r = - 3 \cdot 0 + 3$

$r = 3$

For any value of $x > 4$, you have $x - 4 > 0$ and

$r = {\underbrace{- 3 \cdot \sqrt{x - 4}}}_{\textcolor{b l u e}{< - 3}} + 3 \implies r < 3$

The range of the function will thus be $\left(- \infty , 3\right]$.

graph{-3 * sqrt(x-4) + 3 [-10, 10, -5, 5]}