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 to the questions: Question 1: Arithmetic Mean Given that 17, m, 27 are consecutive terms of an arithmetic progression. Step 1: In an arithmetic progression, the middle term is the arithmetic mean of its neighbors. m = (17 + 27)/(2) Step 2: Calculate the sum and divide. m = (44)/(2) m = 22 The arithmetic mean, m, is 22. Question 2: Inverse Function and Composite Function A function f is defined by f(x) = 2x - 5. Find f^-1(x) Step 1: Let y = f(x), so y = 2x - 5. To find the inverse, swap x and y. x = 2y - 5 Step 2: Solve for y. x + 5 = 2y y = (x + 5)/(2) So, f^-1(x) = (x + 5)/(2). Find f f^-1(x) Step 1: Substitute f^-1(x) into f(x). f f^-1(x) = f((x + 5)/(2)) Step 2: Apply the function f(x) = 2x - 5. f((x + 5)/(2)) = 2((x + 5)/(2)) - 5 f f^-1(x) = (x + 5) - 5 f f^-1(x) = x So, f f^-1(x) = x. Question 3: Value of x if f f(x) = x Given f(x) = 2x - 5. Step 1: Find the composite function f f(x). f f(x) = f(2x - 5) Step 2: Substitute 2x - 5 into f(x). f(2x - 5) = 2(2x - 5) - 5 f f(x) = 4x - 10 - 5 f f(x) = 4x - 15 Step 3: Set f f(x) = x and solve for x. 4x - 15 = x 4x - x = 15 3x = 15 x = (15)/(3) x = 5 The value of x is 5. That's 2 down. 3 left today — send the next one.