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 to the problems. Problem 1: Solving for x Step 1: Divide both sides of the equation by 223. 223x = 87 x = (87)/(223) The value of x is approximately 0.39013. x = (87)/(223) Problem 2: Functions Given F(x) = 2x - 1 and g(x) = x^2 + 1. a) Find g(1+x). Step 1: Substitute (1+x) into the expression for g(x). g(1+x) = (1+x)^2 + 1 Step 2: Expand the expression. g(1+x) = (1+2x+x^2) + 1 g(1+x) = x^2 + 2x + 1 + 1 g(1+x) = x^2 + 2x + 2 g(1+x) = x^2 + 2x + 2 b) Find the range of values of x for which F(x) < -3. Step 1: Set up the inequality using the given function F(x). 2x - 1 < -3 Step 2: Add 1 to both sides of the inequality. 2x < -3 + 1 2x < -2 Step 3: Divide both sides by 2. x < (-2)/(2) x < -1 x < -1 c) Simplify F(x) · g(x). Step 1: Write the product of F(x) and g(x). F(x) · g(x) = (2x - 1)(x^2 + 1) Step 2: Expand the product using the distributive property. F(x) · g(x) = 2x(x^2) + 2x(1) - 1(x^2) - 1(1) F(x) · g(x) = 2x^3 + 2x - x^2 - 1 Step 3: Rearrange the terms in descending order of power. F(x) · g(x) = 2x^3 - x^2 + 2x - 1 F(x) · g(x) = 2x^3 - x^2 + 2x - 1 d) Simplify F(x) - g(x). Step 1: Write the difference F(x) - g(x). F(x) - g(x) = (2x - 1) - (x^2 + 1) Step 2: Distribute the negative sign to the terms in g(x). F(x) - g(x) = 2x - 1 - x^2 - 1 Step 3: Combine like terms and arrange in descending order of power. F(x) - g(x) = -x^2 + 2x - 1 - 1 F(x) - g(x) = -x^2 + 2x - 2 F(x) - g(x) = -x^2 + 2x - 2 Send me the next one 📸