Determine the domain of the argument of the tangent function.

Step 1: Determine the domain of the argument of the tangent function. The function is f(x) = (()/(sqrt(x+1)+4)). For the expression sqrt(x+1) to be defined, we must have x+1 0, which implies x -1. The denominator sqrt(x+1)+4 is always positive since sqrt(x+1) 0, so it will never be zero. Thus, the domain for x is [-1, ). Step 2: Find the range of the inner function, which is the argument of the tangent. Let g(x) = ()/(sqrt(x+1)+4). We need to find the range of g(x) for x [-1, ). • When x = -1: g(-1) = ()/(sqrt(-1+1)+4) = ()/(sqrt(0)+4) = ()/(0+4) = ()/(4) • As x : As x increases, sqrt(x+1) increases, so sqrt(x+1)+4 increases and approaches . Therefore, g(x) = ()/(sqrt(x+1)+4) approaches 0. Since sqrt(x+1)+4 is an increasing function for x -1, the fraction ()/(sqrt(x+1)+4) is a decreasing function. So, the range of the argument = ()/(sqrt(x+1)+4) is (0, ()/(4)]. Step 3: Find the range of f(x) = (). We need to find the range of () for (0, ()/(4)]. The tangent function is strictly increasing on the interval (0, ()/(4)]. • As 0^+: _ 0^+ () = (0) = 0 • When = ()/(4): (()/(4)) = 1 Therefore, the range of f(x) is (0, 1]. Comparing this with the given options: 1. [-1,1] 2. (0,1] 3. [-1,) The correct option is 2. The final answer is (0,1].