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.
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Here are the solutions for the exercise: Given functions: f(x) = 16x + 12 g(x) = (3x + 2)/(6) h(x) = 2x - 2 1. Find f^-1(x) Step 1: Replace f(x) with y. y = 16x + 12 Step 2: Swap x and y. x = 16y + 12 Step 3: Solve for y. x - 12 = 16y y = (x - 12)/(16) Step 4: Replace y with f^-1(x). f^-1(x) = (x - 12)/(16) 2. Find g^-1(x) Step 1: Replace g(x) with y. y = (3x + 2)/(6) Step 2: Swap x and y. x = (3y + 2)/(6) Step 3: Solve for y. 6x = 3y + 2 6x - 2 = 3y y = (6x - 2)/(3) Step 4: Replace y with g^-1(x). g^-1(x) = (6x - 2)/(3) 3. Find h^-1(x) Step 1: Replace h(x) with y. y = 2x - 2 Step 2: Swap x and y. x = 2y - 2 Step 3: Solve for y. x + 2 = 2y y = (x + 2)/(2) Step 4: Replace y with h^-1(x). h^-1(x) = (x + 2)/(2) 4. Find f g, g h, h g Find f g(x) f g(x) = f(g(x)) f(g(x)) = 16((3x + 2)/(6)) + 12 f(g(x)) = (16(3x + 2))/(6) + 12 f(g(x)) = (8(3x + 2))/(3) + 12 f(g(x)) = (24x + 16)/(3) + (36)/(3) f(g(x)) = (24x + 16 + 36)/(3) f g(x) = (24x + 52)/(3) Find g h(x) g h(x) = g(h(x)) g(h(x)) = (3(2x - 2) + 2)/(6) g(h(x)) = (6x - 6 + 2)/(6) g(h(x)) = (6x - 4)/(6) g(h(x)) = (2(3x - 2))/(6) g h(x) = (3x - 2)/(3) Find h g(x) h g(x) = h(g(x)) h(g(x)) = 2((3x + 2)/(6)) - 2 h(g(x)) = (2(3x + 2))/(6) - 2 h(g(x)) = (3x + 2)/(3) - 2 h(g(x)) = (3x + 2)/(3) - (6)/(3) h(g(x)) = (3x + 2 - 6)/(3) h g(x) = (3x - 4)/(3) 5. Find f^-1(g(x)) and h^-1(g(x)) Find f^-1(g(x)) Substitute g(x) = (3x + 2)/(6) into f^-1(x) = (x - 12)/(16). f^-1(g(x)) = ((3x + 2)/(6)) - 1216 f^-1(g(x)) = (3x + 2)/(6) - (72)/(6)16 f^-1(g(x)) = (3x + 2 - 72)/(6)16 f^-1(g(x)) = (3x - 70)/(6 × 16) f^-1(g(x)) = (3x - 70)/(96) Find h^-1(g(x)) Substitute g(x) = (3x + 2)/(6) into h^-1(x) = (x + 2)/(2). h^-1(g(x)) = ((3x + 2)/(6)) + 22 h^-1(g(x)) = (3x + 2)/(6) + (12)/(6)2 h^-1(g(x)) = (3x + 2 + 12)/(6)2 h^-1(g(x)) = (3x + 14)/(6 × 2) h^-1(g(x)) = (3x + 14)/(12)