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 functions P(x) = 2x+1 and g(x) = x-5. a) Find gp(x) Step 1: To find gp(x), we substitute P(x) into g(x). gp(x) = g(P(x)) = g(2x+1) Step 2: Substitute 2x+1 into g(x) = x-5. g(2x+1) = (2x+1) - 5 Step 3: Simplify the expression. g(2x+1) = 2x - 4 The expression for gp(x) is 2x-4. b) Find pg(x) Step 1: To find pg(x), we substitute g(x) into P(x). pg(x) = P(g(x)) = P(x-5) Step 2: Substitute x-5 into P(x) = 2x+1. P(x-5) = 2(x-5) + 1 Step 3: Expand and simplify the expression. P(x-5) = 2x - 10 + 1 P(x-5) = 2x - 9 The expression for pg(x) is 2x-9. c) Find pp(x) Step 1: To find pp(x), we substitute P(x) into P(x). pp(x) = P(P(x)) = P(2x+1) Step 2: Substitute 2x+1 into P(x) = 2x+1. P(2x+1) = 2(2x+1) + 1 Step 3: Expand and simplify the expression. P(2x+1) = 4x + 2 + 1 P(2x+1) = 4x + 3 The expression for pp(x) is 4x+3. d) Find gg(x) Step 1: To find gg(x), we substitute g(x) into g(x). gg(x) = g(g(x)) = g(x-5) Step 2: Substitute x-5 into g(x) = x-5. g(x-5) = (x-5) - 5 Step 3: Simplify the expression. g(x-5) = x - 10 The expression for gg(x) is x-10. That's 2 down. 3 left today — send the next one.