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 are the solutions for the given functions: Given functions: f(x) = 2x + 7 g(x) = (3)/(x-2) a) Find f^-1(x) Step 1: Set y = f(x). y = 2x + 7 Step 2: Swap x and y. x = 2y + 7 Step 3: Solve for y. Subtract 7 from both sides: x - 7 = 2y Divide by 2: y = (x - 7)/(2) Step 4: Replace y with f^-1(x). f^-1(x) = (x - 7)/(2) The inverse function is f^-1(x) = (x - 7)/(2). b) Find gf(x) Step 1: Substitute f(x) into g(x). The definition of gf(x) is g(f(x)). Replace x in g(x) with f(x). gf(x) = g(2x + 7) Step 2: Apply the definition of g(x). gf(x) = (3)/((2x + 7) - 2) Step 3: Simplify the expression. gf(x) = (3)/(2x + 5) The composite function is gf(x) = (3)/(2x + 5). c) Find gf(-2) Step 1: Use the expression for gf(x) found in part b. gf(x) = (3)/(2x + 5) Step 2: Substitute x = -2 into the expression. gf(-2) = (3)/(2(-2) + 5) Step 3: Simplify the denominator. gf(-2) = (3)/(-4 + 5) gf(-2) = (3)/(1) gf(-2) = 3 The value of gf(-2) is 3.