Given that f(x) = 2x - 5 and g(x) = x + 3, find (a) f^-1(x), (b) f^-1(-10), (c) gf(3)

Here are the solutions to the problems. Problem 1: Midpoint and Percentage Error a) The point (5,2) is the midpoint of a straight line joining A(x,9) and B(3,y). Find the value of x and the value of y. Step 1: Use the midpoint formula for the x-coordinate. The midpoint formula is M = ((x_1+x_2)/(2), (y_1+y_2)/(2)). Given midpoint (5,2), A(x,9), and B(3,y). For the x-coordinate: 5 = (x+3)/(2) 10 = x+3 x = 10-3 x = 7 Step 2: Use the midpoint formula for the y-coordinate. For the y-coordinate: 2 = (9+y)/(2) 4 = 9+y y = 4-9 y = -5 The values are x=7 and y=-5. b) The length of line is 8 cm. If this is recorded as 8.2 cm. Calculate the percentage error. Step 1: Calculate the absolute error. Absolute error = | measured value - actual value | Absolute error = |8.2 cm - 8 cm| Absolute error = 0.2 cm Step 2: Calculate the percentage error. Percentage error = Absolute errorActual value × 100\% Percentage error = 0.2 cm8 cm × 100\% Percentage error = 0.025 × 100\% Percentage error = 2.5\% Final Answers for Problem 1: a) x = 7, y = -5 b) The percentage error is 2.5\% --- Problem 2: Functions Given that f(x) = 2x - 5 and g(x) = x + 3, find: a) f^-1(x) Step 1: Set y = f(x) and swap x and y. Let y = 2x - 5. To find the inverse function, 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). b) f^-1(-10) Step 1: Substitute x = -10 into the inverse function f^-1(x). Using f^-1(x) = (x+5)/(2) from part (a): f^-1(-10) = (-10+5)/(2) f^-1(-10) = (-5)/(2) f^-1(-10) = -2.5 c) gf(3) Step 1: First, calculate f(3). Substitute x=3 into f(x) = 2x - 5: f(3) = 2(3) - 5 f(3) = 6 - 5 f(3) = 1 Step 2: Next, substitute the result of f(3) into g(x). Now, calculate g(f(3)), which is g(1). Substitute x=1 into g(x) = x + 3: g(1) = 1 + 3 g(1) = 4 Final Answers for Problem 2: a) f^-1(x) = (x+5)/(2) b) f^-1(-10) = -2.5 c) gf(3) = 4 That's 2 down. 3 left today — send the next one.