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
Step 1: Compute F(-3). The function is defined as: F(x) = x^2 & if x > 1 \\ x - 4 & if x 1 For F(-3), since -3 1, we use the second rule F(x) = x - 4. F(-3) = -3 - 4 = -7 Step 2: Compute F(sqrt(5)). For F(sqrt(5)), we need to compare sqrt(5) with 1. Since sqrt(5) ≈ 2.236, which is greater than 1, we use the first rule F(x) = x^2. F(sqrt(5)) = (sqrt(5))^2 = 5 The computed values are: F(-3) = -7 F(sqrt(5)) = 5 b) Given the function f(x) = -x^2 + 4x - 5. This is a quadratic function in the form ax^2 + bx + c, where a = -1, b = 4, and c = -5. Step 1: Find the line of symmetry. The formula for the line of symmetry of a quadratic function is x = -(b)/(2a). x = -(4)/(2(-1)) = -(4)/(-2) = 2 i) The line of symmetry is x = 2. Step 2: Find the turning point. The x-coordinate of the turning point is the line of symmetry, x = 2. To find the y-coordinate, substitute x = 2 into the function f(x). f(2) = -(2)^2 + 4(2) - 5 f(2) = -4 + 8 - 5 f(2) = 4 - 5 = -1 ii) The turning point is (2, -1). Step 3: Find the maximum or minimum value of f(x). Since the coefficient a = -1 is negative, the parabola opens downwards, which means the turning point is a maximum point. The maximum value of the function is the y-coordinate of the turning point. iii) The maximum value of f(x) is -1.