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 given problems: Problem 1: Matrix Multiplication Step 1: Identify the given matrices. The matrices are given as: B = -2 & 4 C = 3 \\ 4 Step 2: Determine the dimensions of the matrices. Matrix B has 1 row and 2 columns, so its dimension is 1 × 2. Matrix C has 2 rows and 1 column, so its dimension is 2 × 1. Step 3: Check if matrix multiplication BC is possible. For matrix multiplication BC to be defined, the number of columns in B must be equal to the number of rows in C. Number of columns in B = 2. Number of rows in C = 2. Since 2 = 2, the multiplication is possible, and the resulting matrix BC will have dimensions 1 × 1. Step 4: Perform the matrix multiplication. To find the product BC, we multiply the elements of the row of B by the corresponding elements of the column of C and sum them. BC = (-2)(3) + (4)(4) BC = -6 + 16 BC = 10 The final answer is 10 . --- Problem 2: Distance between two points Step 1: Identify the given points. The two points are A = (-12, 2) and B = (-10, 0). Let (x_1, y_1) = (-12, 2) and (x_2, y_2) = (-10, 0). Step 2: Recall the distance formula between two points. The distance d between two points (x_1, y_1) and (x_2, y_2) is given by: d = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2) Step 3: Substitute the coordinates of points A and B into the distance formula. d = sqrt((-10 - (-12))^2 + (0 - 2)^2) Step 4: Simplify the terms inside the parentheses. d = sqrt((-10 + 12)^2 + (-2)^2) d = sqrt((2)^2 + (-2)^2) Step 5: Calculate the squares. d = sqrt(4 + 4) Step 6: Add the terms under the square root. d = sqrt(8) Step 7: Simplify the square root. d = sqrt(4 × 2) d = sqrt(4) × sqrt(2) d = 2sqrt(2) The length of the straight line segment AB is 2sqrt(2). --- Problem 3: Functions Given functions: f(x) = 2x + 1 g(x) = (3)/(x-2) a) Find f^-1(x) Step 1: Set y = f(x). y = 2x + 1 Step 2: Swap x and y. x = 2y + 1 Step 3: Solve for y. Subtract 1 from both sides: x - 1 = 2y Divide by 2: y = (x - 1)/(2) Step 4: Replace y with f^-1(x). f^-1(x) = (x - 1)/(2) The inverse function is f^-1(x) = (x - 1)/(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 + 1) Step 2: Apply the definition of g(x). gf(x) = (3)/((2x + 1) - 2) Step 3: Simplify the expression. gf(x) = (3)/(2x - 1) The composite function is gf(x) = (3)/(2x - 1). c) Find gf(-2) Step 1: Use the expression for gf(x) found in part b. gf(x) = (3)/(2x - 1) Step 2: Substitute x = -2 into the expression. gf(-2) = (3)/(2(-2) - 1) Step 3: Simplify the denominator. gf(-2) = (3)/(-4 - 1) gf(-2) = (3)/(-5) gf(-2) = -(3)/(5) The value of gf(-2) is -(3)/(5).