# What is the range of the function f(x) = -sqrt(x+3)?

Jun 16, 2017

Range : $f \left(x\right) \le 0$, in interval notation: $\left[0 , - \infty\right)$

#### Explanation:

$f \left(x\right) = - \sqrt{x + 3}$ . Output of under root is $\sqrt{x + 3} \ge 0 \therefore f \left(x\right) \le 0$ .

Range : $f \left(x\right) \le 0$ In interval notation: $\left[0 , - \infty\right)$

graph{-(x+3)^0.5 [-10, 10, -5, 5]} [Ans]

Jun 16, 2017

Range: $\left(- \infty , 0\right]$

#### Explanation:

$f \left(x\right) = - \sqrt{x + 3}$

$f \left(x\right) \in \mathbb{R} \forall \left(x + 3\right) \ge 0$

$\therefore f \left(x\right) \in \mathbb{R} \forall x \ge - 3$

$f \left(- 3\right) = 0$ [A]

As $x$ increases beyond all bounds $f \left(x\right) \to - \infty$ [B]

Combining results [A] and [B] the range of $y$ is: $\left(- \infty , 0\right]$

The range of $y$ maybe better understood from the graph of $y$ below.

graph{-sqrt(x+3) [-4.207, 1.953, -2.322, 0.757]}