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.
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Part b) To find the domain D such that the range of f(x) = (1)/(x-2) is [-1, 0). Step 1: Solve y = (1)/(x-2) for x. y = (1)/(x-2) x - 2 = (1)/(y) x = 2 + (1)/(y), y ≠ 0 Step 2: f'(x) = -(1)/((x-2)^2) < 0, so f is strictly decreasing on (-, 2) and on (2, ). On (-, 2), f(x) < 0: as x -, f(x) 0^-; as x 2^-, f(x) -. Step 3: Solve f(x) = -1. (1)/(x-2) = -1 x-2 = -1 x = 1 f(1) = -1. Step 4: Since f is strictly decreasing on (-, 2), on (-, 1], f decreases from 0^- to -1. Thus, f((-, 1]) = [-1, 0). Domain: (-, 1] (-,1] Part c) f(x) = x^2 - 2x. Show f(x) -1 for all real x. Step 1: Complete the square. f(x) = x^2 - 2x = (x^2 - 2x + 1) - 1 = (x-1)^2 - 1 Step 2: (x-1)^2 0 for all real x. f(x) = (x-1)^2 - 1 0 - 1 = -1 Equality at x=1: f(1) = -1. Alternatively, f'(x) = 2x - 2 = 0 gives x=1, f''(x)=2>0 so minimum f(1)=-1. Thus, f(x) -1 for all x. f(x)-1 Part d) Given a^n = b^n, show ( (a)/(b) )^n = 1 (assume b ≠ 0). Step 1: Divide both sides by b^n. a^n = b^n (a^n)/(b^n) = (b^n)/(b^n) ( (a)/(b) )^n = 1 If b=0, then a^n=0 so a=0, but (a)/(b) undefined. **( (a)/(b) )^n = 1