This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
To find the domain of the vector function r(t), we need to find the domain of each of its component functions and then take the intersection of these domains. The given vector function is: r(t) = (t^2 - 1)/(t - 1), sqrt(t+8), (( t))/( t) Let's analyze each component: Step 1: Find the domain of the first component, f_1(t) = (t^2 - 1)/(t - 1). For a rational function, the denominator cannot be zero. t - 1 ≠ 0 t ≠ 1 So, the domain for f_1(t) is D_1 = (-, 1) (1, ). Step 2: Find the domain of the second component, f_2(t) = sqrt(t+8). For a square root function, the expression inside the square root must be non-negative. t + 8 0 t -8 So, the domain for f_2(t) is D_2 = [-8, ). Step 3: Find the domain of the third component, f_3(t) = (( t))/( t). For the natural logarithm t to be defined, its argument must be positive. t > 0 Also, the denominator cannot be zero. t ≠ 0 This implies t ≠ e^0, so t ≠ 1. Combining these conditions, the domain for f_3(t) is D_3 = (0, 1) (1, ). Step 4: Find the intersection of all individual domains. The domain of r(t) is D = D_1 D_2 D_3. D_1 = (-, 1) (1, ) D_2 = [-8, ) D_3 = (0, 1) (1, ) We need t to satisfy all conditions: t ≠ 1, t -8, and t > 0. Combining t -8 and t > 0, we get t > 0. So, the overall conditions are t > 0 and t ≠ 1. This can be written in interval notation as (0, 1) (1, ). The domain of r(t) is: (0, 1) (1, ) Drop the next question.