answer:# Step-by-Step Solution Part (1): Find the value of f(2) - **Step 1:** Let's start with the identities given in the function. Setting x = y = 0 gives us: [f(0 + 0) = f(0) + f(0) - 2] Simplifying, we get: [2f(0) - 2 = f(0)] Solving for f(0), we find that: [f(0) = 2] - **Step 2:** Next, we want to find f(2). Letting x = 2 and y = -2: [f(2 + (-2)) = f(2) + f(-2) - 2] Knowing that f(-2) = 3 and f(0) = 2, we substitute: [2 = f(2) + 3 - 2] Solving for f(2), we find: [f(2) = 1] Hence, the value of f(2) is boxed{1}. Part (2): Determine the monotonicity of the function f(x) - **Proof:** For any m, n in mathbb{R}, if m < n, we analyze f(m) - f(n): [f(m) - f(n) = f((m-n) + n) - f(n) = f(m-n) + f(n) - 2 - f(n)] Simplifying, we get: [f(m) - f(n) = f(m-n) - 2] Since it's given that f(x) > 2 for x < 0, and m-n < 0 by assumption, [f(m-n) > 2] This implies: [f(m) - f(n) > 0] Thus, f(m) > f(n), proving that f(x) is a monotonically decreasing function on mathbb{R}. Hence, f(x)'s monotonicity is boxed{text{monotonically decreasing}}. Part (3): Find the range of real number t - **Step 1:** Given the inequality and simplifying as per the function's properties: [2f(x) - 2 leqslant f[t^{2} + t^{-2} - m(t + t^{-1})] + f(2) - 2] This simplifies to: [f(2x) leqslant f[t^{2} + t^{-2} - m(t + t^{-1}) + 2]] - **Step 2:** Let a = t + t^{-1}. The inequality simplifies to: [2x geqslant a^{2} - ma] Given the range of x in [-3,3] and m in [5,7], the inequality a^{2} - ma leqslant -6 holds. - **Step 3:** Solving the inequalities for m = 5 and m = 7, we find that for both cases, a in [2, 3]. - **Step 4:** Hence, we have: [2 leqslant t + frac{1}{t} leqslant 3] Solving this inequality for t, we find the range of t to be: [t in left[frac{3 - sqrt{5}}{2}, frac{3 + sqrt{5}}{2}right]] Therefore, the range of real number t is boxed{left[frac{3 - sqrt{5}}{2}, frac{3 + sqrt{5}}{2}right]}.

answer:Since 459 div 357 = 1 remainder 102, 357 div 102 = 3 remainder 51, 102 div 51 = 2 with no remainder, Therefore, the greatest common divisor of 459 and 357 is 51. Hence, the answer is boxed{51}.

answer:By definition of the inverse function, we have the relation: y=log_{2}(x+1)+1 To find the original function f(x), we need to express x in terms of y. First, isolate the logarithmic part: y - 1 = log_{2}(x+1) Since log_{2}(x+1) is the exponent to which 2 must be raised to yield (x+1), we can rewrite it as an exponential equation: 2^{y - 1} = x+1 Then, solve for x: x = 2^{y - 1} - 1 Now, to find f(x), we switch x and y: y = 2^{x - 1} - 1 Therefore, the function f(x) is: f(x) = 2^{x - 1} - 1 The process above leads to the conclusion that f(x) = boxed{2^{x - 1} - 1}.

answer:1. **Factor out the leading coefficient**: The leading coefficient of the quadratic term is -4, so factor this out from the x^2 and x terms: -4x^2 + 16x = -4(x^2 - 4x). 2. **Complete the square**: To complete the square inside the parentheses, take half the coefficient of x (which is -4), square it to get 4, and add it inside the square: x^2 - 4x + 4 = (x - 2)^2. Therefore, -4(x^2 - 4x) = -4((x - 2)^2 - 4) = -4(x - 2)^2 + 16. 3. **Match the original polynomial**: Add the constant term from the original polynomial to match it: -4x^2 + 16x + 128 = -4(x - 2)^2 + 16 + 128 = -4(x - 2)^2 + 144. Thus, a = -4, b = -2, and c = 144. Calculate a+b+c: a+b+c = -4 - 2 + 144 = boxed{138}.