6. \( \quad f(x)=\sqrt{2 x+4}, \quad g(x)=\sqrt{2 x+4} \)

It looks like you have two functions defined as follows: \[ f(x) = \sqrt{2x + 4} \] \[ g(x) = \sqrt{2x + 4} \] Since both functions \( f(x) \) and \( g(x) \) are identical, we can analyze them together. ### Domain To find the domain of these functions, we need to ensure that the expression inside the square root is non-negative: \[ 2x + 4 \geq 0 \] Solving this inequality: \[ 2x \geq -4 \\ x \geq -2 \] Thus, the domain of both functions is: \[ [-2, \infty) \] ### Range Next, we can find the range of the functions. The minimum value occurs when \( x = -2 \): \[ f(-2) = \sqrt{2(-2) + 4} = \sqrt{0} = 0 \] As \( x \) increases, \( f(x) \) and \( g(x) \) will also increase without bound. Therefore, the range of both functions is: \[ [0, \infty) \] ### Composition of Functions Since \( f(x) \) and \( g(x) \) are the same, their composition will also yield the same function: \[ f(g(x)) = f(f(x)) = \sqrt{2(\sqrt{2x + 4}) + 4} \] ### Summary - **Domain**: \( [-2, \infty) \) - **Range**: \( [0, \infty) \) - **Composition**: \( f(g(x)) = f(f(x)) = \sqrt{2(\sqrt{2x + 4}) + 4} \) If you have any specific questions or further operations you would like to perform with these functions, please let me know!