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
Here's how to show the given identity: Given the function f(x) = (-2)/(x-1) + 2. Step 1: Find the expression for f((1)/(x)). Substitute x with (1)/(x) in the function f(x): f((1)/(x)) = (-2)/(1)x-1 + 2 Step 2: Simplify f((1)/(x)). Combine the terms in the denominator: f((1)/(x)) = (-2)/(1-x)x + 2 Invert and multiply the fraction: f((1)/(x)) = (-2x)/(1-x) + 2 Step 3: Calculate f((1)/(x)) - f(x). Substitute the expressions for f((1)/(x)) and f(x): f((1)/(x)) - f(x) = ((-2x)/(1-x) + 2) - ((-2)/(x-1) + 2) Step 4: Simplify the expression. Distribute the negative sign and cancel out the constant terms: f((1)/(x)) - f(x) = (-2x)/(1-x) + 2 + (2)/(x-1) - 2 f((1)/(x)) - f(x) = (-2x)/(1-x) + (2)/(x-1) Note that 1-x = -(x-1). Substitute this into the first term: f((1)/(x)) - f(x) = (-2x)/(-(x-1)) + (2)/(x-1) f((1)/(x)) - f(x) = (2x)/(x-1) + (2)/(x-1) Step 5: Combine the fractions. Since the denominators are the same, combine the numerators: f((1)/(x)) - f(x) = (2x+2)/(x-1) Step 6: Factor the numerator. Factor out 2 from the numerator: f((1)/(x)) - f(x) = (2(x+1))/(x-1) This matches the expression we needed to show. That's 2 down. 3 left today — send the next one.