answer:To solve this problem, let's analyze each pair of functions to see if they represent the same function. **For Option A:** fleft(xright) = x - 1 has a domain of all real numbers, mathbb{R}. g(x) = frac{x^2 - 1}{x + 1} can be simplified to g(x) = frac{(x-1)(x+1)}{x+1}, which simplifies to g(x) = x - 1 when x neq -1. Thus, the domain of g(x) is {x | x neq -1}. Since the domains are different (mathbb{R} for f(x) and mathbb{R} setminus {-1} for g(x)), f(x) and g(x) in Option A do not represent the same function. **For Option B:** fleft(xright) = |x + 1|. gleft(xright) is defined piecewise as g(x) = x + 1 for x geq -1 and g(x) = -x - 1 for x < -1. This is precisely how absolute value functions are defined, meaning f(x) and g(x) have the same domain and rule for all x, thus they represent the same function. **For Option C:** fleft(xright) = 1 has a domain of all real numbers, mathbb{R}. gleft(xright) = (x+1)^{0} equals 1 for all x except when x neq -1, as the expression is undefined at x = -1 (since anything to the power of 0 is 1, but the base cannot be a result of division by zero). Hence, the domain is mathbb{R} setminus {-1}. Since the domains are different, f(x) and g(x) in Option C do not represent the same function. **For Option D:** fleft(xright) = x has a domain of all real numbers, mathbb{R}. g(x) = (sqrt{x})^2 is only defined for x geqslant 0 since the square root function requires non-negative inputs. So, the domain of g(x) is {x | x geqslant 0}. The domains differ, meaning f(x) and g(x) in Option D do not represent the same function. **Conclusion:** f(x) and g(x) do not represent the same function in Options A, C, and D. Therefore, the correct answer is boxed{ACD}.

answer:To solve for the number of real solutions to the equation f(x) = 2 based on the piecewise function f(x)=left{begin{array}{ll}x^2+4x+6, & xleq 0 |log_8 x|, & x > 0end{array}right., we proceed as follows: 1. **For x leq 0**, we set the quadratic expression equal to 2: begin{align*} x^2 + 4x + 6 &= 2 x^2 + 4x + 4 &= 0 (x + 2)^2 &= 0. end{align*} This implies that x = -2. 2. **For x > 0**, we work with the absolute value of the logarithmic function: begin{align*} |log_8 x| &= 2. end{align*} This equation can lead to two scenarios due to the absolute value: - log_8 x = 2 - log_8 x = -2 (a) For log_8 x = 2, we translate this into exponential form to find x: begin{align*} 8^2 &= x 64 &= x. end{align*} (b) For log_8 x = -2, we again translate into exponential form: begin{align*} 8^{-2} &= x frac{1}{64} &= x. end{align*} Thus, combining both cases, we find three real solutions for f(x) = 2: x = -2, x = 64, and x = frac{1}{64}. Hence, the total number of real solutions is boxed{3}, corresponding to choice boxed{B}.

answer:1. All excellent students in Daqing Experimental High School cannot form a set, as it does not satisfy the determinacy of a set; 2. 0 is the smallest natural number, hence 0 in mathbb{N}; 3. The set {(x, y) | y = x²} and the set {y | y = x²} do not represent the same set, the former is a set of points, while the latter is a set of numbers; 4. The empty set is a true subset of any non-empty set. From the above, we can conclude that statements 1, 3, and 4 are incorrect; while statement 2 is correct. Therefore, the answer is: boxed{text{A}}. The determinacy of a set can be used to judge statement 1; the fact that 0 is the smallest natural number can be used to judge statement 2; the difference between a set of points and a set of numbers can be used to judge statement 3; the properties of an empty set can be used to judge statement 4. This question tests the concepts of sets and the properties of the empty set, as well as the relationship between elements and sets. It assesses the ability to make judgments and is a basic question.

answer:Let's denote the cost of each mango as M dollars and the cost of each carton of apple juice as A dollars. Rihanna bought 6 mangoes and 6 cartons of apple juice, so the total cost of the mangoes is 6M dollars and the total cost of the apple juice is 6A dollars. The total amount Rihanna spent is the sum of the cost of the mangoes and the apple juice, which is 6M + 6A dollars. Rihanna had 50 initially and has 14 left after shopping, so the amount she spent is 50 - 14 = 36. Therefore, we have the equation: 6M + 6A = 36 Since we have two unknowns (M and A) and only one equation, we cannot determine the individual cost of each mango and each carton of apple juice without additional information. However, we can determine the combined cost of one mango and one carton of apple juice. To find the combined cost of one mango and one carton of apple juice, we can divide the total amount spent by the total number of items (6 mangoes + 6 cartons of apple juice = 12 items): 36 / 12 items = 3 per item So, the combined cost of one mango and one carton of apple juice is boxed{3} . Without further information, we cannot determine the individual costs of a mango and a carton of apple juice.