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
You're on a roll — Step 1: Evaluate g(2). The function g(x) is given by g(x) = x + 1. Substitute x=2 into g(x): g(2) = 2 + 1 = 3 Step 2: Express f(x + (1)/(x)) in a simpler form. The function f is defined by f(x + (1)/(x)) = x^2 + (1)/(x^2). We know the algebraic identity (a+b)^2 = a^2 + 2ab + b^2. Let a=x and b=(1)/(x). Then: (x + (1)/(x))^2 = x^2 + 2 · x · (1)/(x) + ((1)/(x))^2 (x + (1)/(x))^2 = x^2 + 2 + (1)/(x^2) Rearranging this, we get: x^2 + (1)/(x^2) = (x + (1)/(x))^2 - 2 So, we can rewrite the definition of f as: f(x + (1)/(x)) = (x + (1)/(x))^2 - 2 Let u = x + (1)/(x). Then the function f can be written as f(u) = u^2 - 2. Step 3: Calculate f(g(2)). From Step 1, we found g(2) = 3. Now we need to find f(3). Using the simplified form of f(u) from Step 2: f(3) = (3)^2 - 2 f(3) = 9 - 2 f(3) = 7 Step 4: Select the correct option. The calculated value is 7, which corresponds to option (B). The final answer is B 4 down, 1 to go! What's next?