Let's analyze the equation $y = \sqrt{x + 4} - 3$ to determine its domain and range.

Domain:

• The square root function $\sqrt{x+4}$ is defined only when the expression inside the square root is non-negative. Therefore, $x + 4 \geq 0$, which simplifies to $x \geq -4$.
• So, the domain is $x \geq -4$.

Range:

• When $x = -4$, the square root part $\sqrt{x + 4}$ becomes $\sqrt{0} = 0$, and the equation becomes $y = 0 - 3 = -3$.
• As $x$ increases, $\sqrt{x + 4}$ becomes positive, making $y = \sqrt{x + 4} - 3$ larger than $-3$.
• Therefore, the smallest value of $y$ is $-3$, and as $x$ increases, $y$ increases without bound.
• So, the range is $y \geq -3$.

Conclusion:

The correct answer is:

The domain of the equation is " x is greater than or equal to -4" and the range is " y is greater than or equal to -3".

This matches the third option:

Domain: $x \geq -4$
Range: $y \geq -3$

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Would you like more details or further explanation?

Here are 5 related questions:

1. How do you find the domain of a function involving square roots?
2. Can the range of a square root function ever include negative values?
3. How does adding or subtracting a constant outside the square root affect the range?
4. What would happen to the domain and range if the function were $y = -\sqrt{x + 4} - 3$?
5. How do transformations like shifting and reflecting impact the domain and range of functions?

Tip: For functions involving square roots, always check the inside of the root for non-negative values to determine the domain!