To determine the domain and range of the function $y = \sqrt{x + 4} - 3$, we need to analyze the function's behavior, particularly the expression inside the square root and the overall output of the function.

Step 1: Determine the Domain

The domain of a function includes all values of $x$ for which the function is defined. Since there is a square root in the function, the expression inside the square root must be non-negative (greater than or equal to zero).

$x + 4 \geq 0$

Solving for $x$:

$x \geq -4$

Thus, the domain of the function is all $x$ values such that $x \geq -4$.

Domain: $x \in [-4, \infty)$

Step 2: Determine the Range

The range of the function is the set of all possible output values $y$.

The function is $y = \sqrt{x + 4} - 3$. Let's examine it:

1. The square root function $\sqrt{x + 4}$ produces values that are $\geq 0$ since a square root cannot be negative.
2. Therefore, $\sqrt{x + 4}$ will range from 0 to $\infty$ as $x$ increases from $-4$ to $\infty$.
3. Subtracting 3 from $\sqrt{x + 4}$ shifts the entire range down by 3 units.

For $x = -4$: $y = \sqrt{-4 + 4} - 3 = \sqrt{0} - 3 = -3$

As $x$ increases, $y$ will increase without bound.

Range: $y \in [-3, \infty)$

Summary:

• Domain: $x \geq -4$ or $x \in [-4, \infty)$
• Range: $y \geq -3$ or $y \in [-3, \infty)$

Would you like to explore this further or ask another question?

Related Questions:

1. How does the graph of $y = \sqrt{x + 4} - 3$ compare to $y = \sqrt{x}$?
2. How do transformations affect the domain and range of a function?
3. What is the general approach for finding the domain of a square root function?
4. Can the range of a function ever include negative infinity?
5. How would the function change if it were $y = -\sqrt{x + 4} - 3$?

Tip:

When analyzing functions, start by addressing the domain restrictions first (e.g., avoiding negative square roots, zeros in denominators) before moving to the range, which depends on the outputs generated by valid domain values.