5. \( f(x)=(x-4)^{2},[4, \infty) \) 6. \( f(x)=(x+2)^{2},[-2, \infty) \) 7. \( f(x)=(x+1)^{2}-3, \quad[-1, \infty) \) 8. \( f(x)=2-\sqrt{3+x} \)

It looks like you have provided a list of functions along with their domains. Let's analyze each function one by one. ### 5. \( f(x) = (x-4)^{2}, \quad [4, \infty) \) - **Function Description**: This is a quadratic function that opens upwards, with its vertex at \( x = 4 \). - **Domain**: The function is defined for \( x \) values starting from 4 and going to infinity. - **Range**: Since the vertex is at \( (4, 0) \) and the function opens upwards, the range is \( [0, \infty) \). ### 6. \( f(x) = (x+2)^{2}, \quad [-2, \infty) \) - **Function Description**: This is also a quadratic function that opens upwards, with its vertex at \( x = -2 \). - **Domain**: The function is defined for \( x \) values starting from -2 and going to infinity. - **Range**: The vertex is at \( (-2, 0) \), so the range is \( [0, \infty) \). ### 7. \( f(x) = (x+1)^{2} - 3, \quad [-1, \infty) \) - **Function Description**: This is a quadratic function that opens upwards, with its vertex at \( x = -1 \). - **Domain**: The function is defined for \( x \) values starting from -1 and going to infinity. - **Range**: The vertex is at \( (-1, -3) \), so the range is \( [-3, \infty) \). ### 8. \( f(x) = 2 - \sqrt{3+x} \) - **Function Description**: This function involves a square root, which means we need to consider the domain where the expression under the square root is non-negative. - **Domain**: The expression \( 3 + x \geq 0 \) gives \( x \geq -3 \). Therefore, the domain is \( [-3, \infty) \). - **Range**: As \( x \) increases from -3 to infinity, \( \sqrt{3+x} \) increases from 0 to infinity, making \( f(x) \) decrease from \( 2 - 0 = 2 \) to \( 2 - \infty = -\infty \). Thus, the range is \( (-\infty, 2] \). ### Summary of Functions 1. \( f(x) = (x-4)^{2}, \quad [4, \infty) \) - Range: \( [0, \infty) \) 2. \( f(x) = (x+2)^{2}, \quad [-2, \infty) \) - Range: \( [0, \infty) \) 3. \( f(x) = (x+1)^{2} - 3, \quad [-1, \infty) \) - Range: \( [-3, \infty) \) 4. \( f(x) = 2 - \sqrt{3+x} \) - Domain: \( [-3, \infty) \) - Range: \( (-\infty, 2] \) If you have any specific questions or need further analysis on any of these functions, feel free to ask!