Use graphing technology to find the domain of the function \( f(x)=\sqrt{x+3}+4 \)

To determine the domain of the function \( f(x) = \sqrt{x + 3} + 4 \) using graphing technology, follow these steps: ### 1. Understand the Function The function consists of a square root and a constant: - **Square Root Part:** \( \sqrt{x + 3} \) - **Constant Part:** \( +4 \) ### 2. Determine the Domain Restrictions The expression inside the square root, \( x + 3 \), must be **non-negative** because the square root of a negative number is not a real number. \[ x + 3 \geq 0 \] \[ x \geq -3 \] ### 3. Graph the Function Using graphing technology (like Desmos, GeoGebra, or a graphing calculator): 1. **Plot the Function:** Enter \( f(x) = \sqrt{x + 3} + 4 \) into the graphing tool. 2. **Observe the Graph:** - The graph will start at the point where \( x = -3 \). - For \( x < -3 \), the function is not defined (there are no real values). - For \( x \geq -3 \), the graph will show the behavior of the square root function shifted horizontally and vertically. ### 4. Identify the Domain from the Graph By examining the graph: - The function begins at \( x = -3 \) and extends to the right. - There are no breaks or gaps beyond \( x = -3 \). ### 5. **Conclusion: The Domain** Based on both the algebraic restriction and the graph: \[ \text{Domain of } f(x) = \sqrt{x + 3} + 4 \text{ is } x \geq -3 \] **In interval notation:** \[ \boxed{\,[-3,\ \infty) \,} \]