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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Hey Dany — ready when you are. 4. (i) (a) Step 1: Assume f(x_1) = f(x_2) for x_1, x_2 R, x_1 ≠ 1, x_2 ≠ 1. (4x_1)/(x_1-1) = (4x_2)/(x_2-1) Step 2: Simplify the equation. Divide both sides by 4: (x_1)/(x_1-1) = (x_2)/(x_2-1) Cross-multiply: x_1(x_2-1) = x_2(x_1-1) x_1x_2 - x_1 = x_1x_2 - x_2 Step 3: Isolate x_1 and x_2. Subtract x_1x_2 from both sides: -x_1 = -x_2 Multiply by -1: x_1 = x_2 Since f(x_1) = f(x_2) implies x_1 = x_2, the function f is injective. 4. (i) (b) Step 1: Find the composite function (f f)(x). (f f)(x) = f(f(x)) = f((4x)/(x-1)) Substitute (4x)/(x-1) into the function f(x) = (4x)/(x-1): (f f)(x) = (4(4x)/(x-1))((4x)/(x-1))-1 Step 2: Simplify the expression. = (16x)/(x-1)(4x - (x-1))/(x-1) = (16x)/(x-1)(4x - x + 1)/(x-1) = (16x)/(x-1)(3x + 1)/(x-1) = (16x)/(x-1) · (x-1)/(3x+1) (f f)(x) = (16x)/(3x+1) Step 3: Determine the domain of (f f)(x). For (f f)(x) to be defined, two conditions must be met: 1. x must be in the domain of f. This means x ≠ 1. 2. f(x) must be in the domain of f. This means f(x) ≠ 1. (4x)/(x-1) ≠ 1 4x ≠ x-1 3x ≠ -1 x ≠ -(1)/(3) Also, the denominator of the simplified (f f)(x) cannot be zero, so 3x+1 ≠ 0 x ≠ -(1)/(3). Combining these conditions, the domain is x R, x ≠ 1, x ≠ -(1)/(3). 4. (ii) (a) The statement p q means "p and q". p: The principal goes to the ministry. q: The principal will meet the minister. Therefore, p q in ordinary English is: The principal goes to the ministry and will meet the minister. What's next?