# What is the domain and range of y = - sqrt(9-x^2)?

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Aug 21, 2015

Domain: $\left[- 3 , 3\right]$
Range: $\left[- 3 , 0\right]$

#### Explanation:

In order to find the function's domain, you need to take into account the fact that, for real numbers, you can only take the square root of a positive number.

In other words, in oerder for the function to be defined, you need the expression that's under the square root to be positive.

$9 - {x}^{2} \ge 0$

${x}^{2} \le 9 \implies | x | \le 3$

This means that you have

$x \ge - 3 \text{ }$ and $\text{ } x \le 3$

For any value of $x$ outside the interval $\left[- 3 , 3\right]$, the expression under the square root will be negative, which means that the function will be undefined. Therefore, the domain of the function will be $x \in \left[- 3 , 3\right]$.

Now for the range. For any value of $x \in \left[- 3 , 3\right]$, the function will be negative.

The maximum value the expression under the radical can take is for $x = 0$

$9 - {0}^{2} = 9$

which means that the minimum value of the function will be

$y = - \sqrt{9} = - 3$

Therefore, the range of the function will be $\left[- 3 , 0\right]$.

graph{-sqrt(9-x^2) [-10, 10, -5, 5]}

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