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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11. If f(x) = 2x + 3, then find the value of f f(-1). Step 1: Calculate f(-1). Substitute x = -1 into the function f(x) = 2x + 3. f(-1) = 2(-1) + 3 f(-1) = -2 + 3 f(-1) = 1 Step 2: Calculate f f(-1), which is f(f(-1)). From Step 1, we know f(-1) = 1. So, we need to find f(1). Substitute x = 1 into the function f(x) = 2x + 3. f(1) = 2(1) + 3 f(1) = 2 + 3 f(1) = 5 The value of f f(-1) is 5. 12. Find the vertex of the parabola formed from the quadratic equation y = x^2 - 5x + 6. The given quadratic equation is y = x^2 - 5x + 6. This is in the standard form y = ax^2 + bx + c, where a=1, b=-5, and c=6. Step 1: Find the x-coordinate of the vertex. The x-coordinate of the vertex is given by the formula x = -(b)/(2a). x = -(-5)/(2(1)) x = (5)/(2) Step 2: Find the y-coordinate of the vertex. Substitute the x-coordinate (x = (5)/(2)) back into the quadratic equation y = x^2 - 5x + 6. y = ((5)/(2))^2 - 5((5)/(2)) + 6 y = (25)/(4) - (25)/(2) + 6 To combine these terms, find a common denominator, which is 4. y = (25)/(4) - (25 × 2)/(2 × 2) + (6 × 4)/(1 × 4) y = (25)/(4) - (50)/(4) + (24)/(4) y = (25 - 50 + 24)/(4) y = (-25 + 24)/(4) y = (-1)/(4) The vertex of the parabola is (x, y) = ((5)/(2), -(1)/(4)). The vertex of the parabola is ((5)/(2), -(1)/(4)).