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
Step 1: Find the inverse function h^-1(x). The function h(x) is given by h(x) = (1)/(x+2). To find the inverse, let y = h(x), so y = (1)/(x+2). Swap x and y: x = (1)/(y+2) Now, solve for y: x(y+2) = 1 xy + 2x = 1 xy = 1 - 2x y = (1 - 2x)/(x) Thus, the inverse function is: h^-1(x) = (1 - 2x)/(x), x ≠ 0 Step 2: Calculate h(-(1)/(2)). The function h(x) is h(x) = (1)/(x+2). Substitute x = -(1)/(2) into h(x): h(-(1)/(2)) = (1)/(-1)2 + 2 To simplify the denominator, find a common denominator: -(1)/(2) + 2 = -(1)/(2) + (4)/(2) = (-1+4)/(2) = (3)/(2) So, h(-(1)/(2)) = (1)/(3)2 h(-(1)/(2)) = (2)/(3) Step 3: Calculate g h (-(1)/(2)). We need to find g(h(-(1)/(2))). From Step 2, we know h(-(1)/(2)) = (2)/(3). The function g(x) is given by g(x) = x^2. Substitute x = (2)/(3) into g(x): g((2)/(3)) = ((2)/(3))^2 g((2)/(3)) = (2^2)/(3^2) g((2)/(3)) = (4)/(9) Therefore, g h (-(1)/(2)) = (4)/(9) 3 done, 2 left today. You're making progress.