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
Given the functions: f(x) = sqrt(2-x) g(x) = x^2 + 5 Step 1: Calculate f g(x). The composite function f g(x) is defined as f(g(x)). We substitute g(x) into f(x). f(g(x)) = f(x^2 + 5) Now, replace x in f(x) with (x^2 + 5): f(x^2 + 5) = sqrt(2 - (x^2 + 5)) Simplify the expression inside the square root: f(x^2 + 5) = sqrt(2 - x^2 - 5) f(x^2 + 5) = sqrt(-x^2 - 3) The result for f g(x) is sqrt(-x^2 - 3). Step 2: Calculate g f(x). The composite function g f(x) is defined as g(f(x)). We substitute f(x) into g(x). g(f(x)) = g(sqrt(2-x)) Now, replace x in g(x) with (sqrt(2-x)): g(sqrt(2-x)) = (sqrt(2-x))^2 + 5 Simplify the expression. Note that (sqrt(A))^2 = A for A 0: g(sqrt(2-x)) = (2-x) + 5 g(sqrt(2-x)) = 7-x The result for g f(x) is 7-x. Drop the next question.