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
Right C, let's go. Step 1: Understand the given functions and the composition. We are given two functions: g(x) = x+5 And the composite function: g(f(x)) = (6(x-3))/(x-4) The notation g(f(x)) means that we substitute f(x) into the function g(x) wherever x appears. Step 2: Express g(f(x)) in terms of f(x). Using the definition of g(x), replace x with f(x): g(f(x)) = f(x) + 5 Step 3: Set up an equation to solve for f(x). Now we have two expressions for g(f(x)), so we can set them equal to each other: f(x) + 5 = (6(x-3))/(x-4) Step 4: Isolate f(x). Subtract 5 from both sides of the equation: f(x) = (6(x-3))/(x-4) - 5 Step 5: Simplify the expression for f(x). To combine the terms, find a common denominator, which is (x-4): f(x) = (6(x-3))/(x-4) - (5(x-4))/(x-4) Combine the numerators: f(x) = (6(x-3) - 5(x-4))/(x-4) Expand the terms in the numerator: f(x) = (6x - 18 - (5x - 20))/(x-4) Distribute the negative sign: f(x) = (6x - 18 - 5x + 20)/(x-4) Combine like terms in the numerator: f(x) = ((6x - 5x) + (-18 + 20))/(x-4) f(x) = (x + 2)/(x-4) The value of f(x) is: f(x) = (x+2)/(x-4) What's next? 🤔