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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You're on a roll — 4.1 The equation of the parabola is f(x) = (x-4)^2 - 9. This is in the form y = a(x-p)^2 + q, where (p, q) is the turning point. Comparing the given equation, the turning point E is (4, -9). The coordinates of E are (4, -9). 4.2 A is an x-intercept of f(x). To find the x-intercepts, set f(x) = 0. (x-4)^2 - 9 = 0 (x-4)^2 = 9 Take the square root of both sides: x-4 = ±sqrt(9) x-4 = ± 3 Solve for x: x-4 = 3 x = 7 x-4 = -3 x = 1 From the graph, A is the x-intercept with the smaller x-value. The coordinates of A are (1, 0). 4.3 First, find the coordinates of C. C is the y-intercept of f(x). Set x=0. f(0) = (0-4)^2 - 9 f(0) = (-4)^2 - 9 f(0) = 16 - 9 f(0) = 7 So, C is (0, 7). The axis of symmetry of f(x) = (x-4)^2 - 9 is the vertical line x=4 (the x-coordinate of the turning point). M is the reflection of C(0, 7) in the line x=4. The x-coordinate of C is 0. The distance from C to the axis of symmetry (x=4) is 4-0=4 units. The x-coordinate of M will be 4+4=8. The y-coordinate remains the same during reflection across a vertical line. The coordinates of M are (8, 7). 4.4 The equation of line g is in the form y = mx+c. C is the y-intercept of f and g. From 4.3, C is (0, 7). So, c=7. The equation of g is y = mx+7. D is the x-intercept of g. DE is parallel to the y-axis, which means D and E have the same x-coordinate. From 4.1, E is (4, -9). So, the x-coordinate of D is 4. Since D is an x-intercept, its y-coordinate is 0. So, D is (4, 0). Now, calculate the gradient m using points C(0, 7) and D(4, 0): m = (y_2 - y_1)/(x_2 - x_1) = (0 - 7)/(4 - 0) = (-7)/(4) The equation of g is y = -(7)/(4)x + 7. 4.5 To find the inverse function g^-1, swap x and y in the equation for g and solve for y. The equation of g is y = -(7)/(4)x + 7. Swap x and y: x = -(7)/(4)y + 7 Subtract 7 from both sides: x - 7 = -(7)/(4)y Multiply by -(4)/(7): -(4)/(7)(x - 7) = y y = -(4)/(7)x + (-(4)/(7))(-7) y = -(4)/(7)x + 4 The equation of g^-1 is y = -(4)/(7)x + 4. 4.6 We need to find the values of x for which x(f(x)) 0. This inequality holds true if: 1. x 0 AND f(x) 0 2. x 0 AND f(x) 0 From 4.2, the x-intercepts of f(x) are x=1 (point A) and x=7 (point B). Since f(x) is an upward-opening parabola, f(x) 0 when x is between or equal to the x-intercepts. So, f(x) 0 for 1 x 7. And f(x) 0 for x 1 or x 7. Now, let's apply these to the two cases: Case 1: x 0 AND f(x) 0 This means x [0, ) AND x [1, 7]. The intersection of these two conditions is 1 x 7. Case 2: x 0 AND f(x) 0 This means x (-, 0] AND (x (-, 1] or x [7, )). The intersection of these two conditions is x 0. Combining the results from Case 1 and Case 2, the values of x for which x(f(x)) 0 are: x 0 or 1 x 7. Send me the next one 📸