answer：Okay, so I have this function f(x) that's defined piecewise. It means it has different expressions depending on whether x is greater than or equal to zero or less than zero. The function is given as:f(x) = {2^x - 1, when x ≥ 0- x² - 2x, when x < 0}And I need to find the real number a such that f(a) = 1. Hmm, okay. So, I guess I need to solve the equation f(a) = 1, which means I have to consider both cases of the piecewise function.First, let me recall what a piecewise function is. It's a function that is defined by different expressions over different intervals of the independent variable—in this case, x. So, depending on whether a is non-negative or negative, I'll use the corresponding expression for f(a).Alright, so let's break it down into two cases:**Case 1: a ≥ 0**In this case, f(a) = 2^a - 1. I need to set this equal to 1 and solve for a.So, 2^a - 1 = 1Let me solve this equation step by step.First, add 1 to both sides:2^a - 1 + 1 = 1 + 1Which simplifies to:2^a = 2Okay, so 2 raised to what power gives me 2? Well, 2^1 = 2, so a = 1.Wait, but let me make sure I didn't skip any steps. So, 2^a = 2. Since the bases are the same, I can set the exponents equal. So, a = 1. That makes sense.So, in this case, a = 1 is a solution.**Case 2: a < 0**Here, f(a) = -a² - 2a. I need to set this equal to 1 and solve for a.So, -a² - 2a = 1Hmm, this is a quadratic equation. Let me rearrange it to standard quadratic form.First, I'll move all terms to one side:- a² - 2a - 1 = 0Wait, actually, let me write it as:- a² - 2a = 1Then, subtract 1 from both sides:- a² - 2a - 1 = 0Alternatively, I can multiply both sides by -1 to make the coefficient of a² positive, which might make it easier to solve.Multiplying both sides by -1:a² + 2a + 1 = 0Now, this looks like a perfect square trinomial. Let me check:a² + 2a + 1 = (a + 1)²Yes, that's correct. So, the equation becomes:(a + 1)² = 0Taking the square root of both sides:a + 1 = 0Therefore, a = -1Alright, so in this case, a = -1 is another solution.Wait, but I need to make sure that this solution fits the condition for this case, which is a < 0. Since -1 is indeed less than 0, this solution is valid.So, putting it all together, the solutions are a = 1 and a = -1.Let me double-check both solutions to make sure they satisfy the original equation.**Checking a = 1:**Since a = 1 is greater than or equal to 0, we use f(a) = 2^a - 1.f(1) = 2^1 - 1 = 2 - 1 = 1Which matches the given condition f(a) = 1. So, a = 1 is correct.**Checking a = -1:**Since a = -1 is less than 0, we use f(a) = -a² - 2a.f(-1) = -(-1)^2 - 2*(-1) = -1 + 2 = 1Again, this matches f(a) = 1. So, a = -1 is also correct.Therefore, both solutions are valid.I think that's all there is to it. I considered both cases of the piecewise function, solved the resulting equations, and verified the solutions. So, the real numbers a that satisfy f(a) = 1 are 1 and -1.**Final Answer**The value of the real number ( a ) is boxed{pm 1}.

answer：Okay, so I have this problem here about calculating the percentage of cost for overhead. Let me read it again to make sure I understand what's being asked."The purchase price of an article is 48. In order to include a certain percentage of cost for overhead and to provide 12 of net profit, the markup should be 30. What is the percentage of cost for overhead?"Alright, so the article costs 48 to purchase. The seller wants to add a markup to cover overhead and also make a net profit of 12. The total markup needed is 30. I need to find out what percentage of the purchase price this overhead represents.First, let me break down the components involved:1. **Purchase Price**: This is the cost price of the article, which is 48.2. **Markup**: This is the amount added to the purchase price to cover overhead and profit. The total markup is 30.3. **Net Profit**: This is the desired profit after covering all costs, which is 12.4. **Overhead**: This is the cost that needs to be covered by the markup, excluding the net profit.So, the markup is used to cover both the overhead and the net profit. That means the markup can be broken down into two parts: one part for overhead and another part for profit.Let me represent this mathematically. Let's denote:- ( P ) = Purchase Price = 48- ( M ) = Markup = 30- ( N ) = Net Profit = 12- ( O ) = OverheadFrom the problem, we know that the total markup ( M ) is the sum of the overhead ( O ) and the net profit ( N ). So,[ M = O + N ]We can rearrange this to solve for ( O ):[ O = M - N ]Plugging in the known values:[ O = 30 - 12 = 18 ]So, the overhead is 18.Now, we need to find the percentage of the purchase price that this overhead represents. To find the percentage, we can use the formula:[ text{Percentage} = left( frac{text{Overhead}}{text{Purchase Price}} right) times 100 ]Substituting the values we have:[ text{Percentage} = left( frac{18}{48} right) times 100 ]Let me compute this step by step. First, divide 18 by 48:[ frac{18}{48} = 0.375 ]Then, multiply by 100 to get the percentage:[ 0.375 times 100 = 37.5% ]So, the overhead is 37.5% of the purchase price.Wait a second, let me double-check my calculations to make sure I didn't make a mistake.Starting again:Markup = Overhead + Net Profit30 = Overhead + 12Subtracting 12 from both sides:Overhead = 30 - 12 = 18That seems correct.Now, to find the percentage of the purchase price that 18 represents:[ frac{18}{48} = 0.375 ]Multiplying by 100 gives 37.5%.Yes, that seems consistent.Just to ensure I'm not missing anything, let me think about the total selling price.Total Selling Price = Purchase Price + Markup[ 48 + 30 = 78 ]So, the selling price is 78.If the selling price is 78, and the net profit is 12, then the total cost (purchase price plus overhead) should be:Total Cost = Selling Price - Net Profit[ 78 - 12 = 66 ]Wait, that doesn't seem right because the purchase price is already 48, and if overhead is 18, then total cost is:Total Cost = Purchase Price + Overhead[ 48 + 18 = 66 ]Which matches the previous calculation. So, that checks out.Therefore, the overhead is indeed 18, which is 37.5% of the purchase price.I think I've got it right. The percentage of cost for overhead is 37.5%.

answer：Okay, so I have this triangle ABC, and I know that the sides opposite angles A and B are a and b respectively. The problem says that a equals 3 times b times the sine of A, which is written as a = 3b sin A. I need to find sin B. Hmm, okay, let me think about how to approach this.First, I remember that in any triangle, the Law of Sines is a useful tool. The Law of Sines states that a/sin A = b/sin B = c/sin C, where a, b, c are the sides opposite angles A, B, C respectively. So, maybe I can use this relationship to find sin B.Given that a = 3b sin A, I can substitute this into the Law of Sines equation. Let me write that down:a / sin A = b / sin BBut since a is equal to 3b sin A, I can replace a with 3b sin A:(3b sin A) / sin A = b / sin BWait, that looks promising. The sin A in the numerator and denominator on the left side will cancel out, right? So, simplifying that:3b = b / sin BNow, I have 3b equals b divided by sin B. Let me write that as:3b = b / sin BHmm, I can divide both sides of this equation by b, assuming b is not zero, which makes sense because b is a side length of a triangle and can't be zero. So, dividing both sides by b:3 = 1 / sin BThat simplifies things. So, 3 equals 1 over sin B. To solve for sin B, I can take the reciprocal of both sides:sin B = 1 / 3So, sin B is equal to one-third. Let me just double-check my steps to make sure I didn't make a mistake.1. I started with the Law of Sines: a / sin A = b / sin B.2. Substituted a with 3b sin A: (3b sin A) / sin A = b / sin B.3. Simplified the left side by canceling sin A: 3b = b / sin B.4. Divided both sides by b: 3 = 1 / sin B.5. Took the reciprocal: sin B = 1/3.Everything seems to check out. I don't see any errors in my reasoning. So, I think the answer is correct.Just to make sure, let me think about another way to approach this problem. Maybe using the Law of Cosines? But I don't think that's necessary here because the Law of Sines directly relates the sides and the sines of their opposite angles, which is exactly what we need.Alternatively, could I have used the area formula for a triangle? The area can be expressed as (1/2)ab sin C, but I don't think that would help here since we don't have information about the area or other sides or angles.No, the Law of Sines seems to be the right approach here. So, I feel confident that sin B is indeed 1/3.**Final Answer**The value of sin B is boxed{dfrac{1}{3}}.

answer：Okay, so I need to solve this problem about a fruit store. Let me read it again to make sure I understand.The store sells a type of fruit with a cost price of 40 yuan per kilogram. The weekly sales quantity, y (in kilograms), is a linear function of the selling price, x (in yuan per kilogram). There's a table with some data points:When x is 45, y is 110;x is 60, y is 80;x is 70, y is 60;x is 75, y is 50.There are three parts to the problem:1. Find the function expression between y and x.2. Determine the selling price that maximizes weekly profit and find that maximum profit.3. If the cost price increases by m yuan per kilogram, and it's observed that when the selling price is above 76 yuan, the weekly profit decreases as the price increases, find the range of m.Alright, let's tackle each part step by step.**Part 1: Finding the function between y and x**Since y is a linear function of x, it should have the form y = kx + b, where k is the slope and b is the y-intercept.We can use two points from the table to find k and b. Let's pick x=60, y=80 and x=70, y=60.First, let's find the slope k.The formula for slope is (y2 - y1)/(x2 - x1).So, k = (60 - 80)/(70 - 60) = (-20)/10 = -2.So, the slope k is -2.Now, let's find b using one of the points. Let's use x=60, y=80.Plug into y = kx + b:80 = (-2)(60) + b80 = -120 + bb = 80 + 120b = 200.So, the linear function is y = -2x + 200.Wait, let me check with another point to make sure.Let's use x=45, y=110.y = -2(45) + 200 = -90 + 200 = 110. Perfect.And x=75, y=50.y = -2(75) + 200 = -150 + 200 = 50. Correct.So, part 1 is done. The function is y = -2x + 200.**Part 2: Maximizing weekly profit**Profit is calculated as (Selling Price - Cost Price) * Quantity Sold.Given that the cost price is 40 yuan per kg, the profit per kg is (x - 40). The quantity sold is y = -2x + 200.So, the total profit, let's call it P, is:P = (x - 40) * yP = (x - 40)(-2x + 200)Let me expand this:P = (x - 40)(-2x + 200)= x*(-2x + 200) - 40*(-2x + 200)= -2x² + 200x + 80x - 8000= -2x² + 280x - 8000So, P = -2x² + 280x - 8000.This is a quadratic equation in terms of x, and since the coefficient of x² is negative (-2), the parabola opens downward, meaning the vertex is the maximum point.The vertex of a parabola given by ax² + bx + c is at x = -b/(2a).Here, a = -2, b = 280.So, x = -280/(2*(-2)) = -280/(-4) = 70.So, the selling price that maximizes profit is 70 yuan per kg.Now, let's find the maximum profit by plugging x=70 into P.P = -2*(70)² + 280*(70) - 8000= -2*4900 + 19600 - 8000= -9800 + 19600 - 8000= (19600 - 9800) - 8000= 9800 - 8000= 1800.So, the maximum profit is 1800 yuan per week.Wait, let me double-check the calculations.Alternatively, since we know the vertex form of a quadratic is P = a(x - h)² + k, where (h, k) is the vertex.Given that P = -2x² + 280x - 8000, we can complete the square.Factor out -2 from the first two terms:P = -2(x² - 140x) - 8000.Now, to complete the square inside the parentheses:Take half of 140, which is 70, square it: 70² = 4900.Add and subtract 4900 inside the parentheses:P = -2[(x² - 140x + 4900 - 4900)] - 8000= -2[(x - 70)² - 4900] - 8000= -2(x - 70)² + 9800 - 8000= -2(x - 70)² + 1800.Yes, so the vertex is at (70, 1800), confirming that maximum profit is 1800 yuan at x=70.**Part 3: Determining the range of m**Now, the cost price increases by m yuan per kg, so the new cost price is (40 + m) yuan per kg.The profit function becomes:P = (x - (40 + m)) * y= (x - 40 - m)(-2x + 200)Let me expand this:P = (x - 40 - m)(-2x + 200)= x*(-2x + 200) - (40 + m)*(-2x + 200)= -2x² + 200x + 2(40 + m)x - 200(40 + m)= -2x² + 200x + 80x + 2mx - 8000 - 200m= -2x² + (200 + 80 + 2m)x - (8000 + 200m)= -2x² + (280 + 2m)x - (8000 + 200m)So, P = -2x² + (280 + 2m)x - (8000 + 200m)Again, this is a quadratic in x, opening downward, so the maximum is at the vertex.The x-coordinate of the vertex is at x = -b/(2a).Here, a = -2, b = (280 + 2m).So, x = -(280 + 2m)/(2*(-2)) = -(280 + 2m)/(-4) = (280 + 2m)/4 = (140 + m)/2.So, the selling price that maximizes profit is x = (140 + m)/2.But the problem states that when the selling price is greater than 76 yuan, the weekly profit decreases as the selling price increases. That means that the vertex must be at x ≤ 76, because beyond that point, the profit decreases.So, (140 + m)/2 ≤ 76Multiply both sides by 2:140 + m ≤ 152Subtract 140:m ≤ 12Since m > 0, as given, the range of m is 0 < m ≤ 12.Wait, let me verify.If m = 12, then the vertex is at (140 + 12)/2 = 152/2 = 76. So, at m=12, the maximum profit is at x=76. Beyond that, profit decreases.If m were greater than 12, say m=13, then the vertex would be at (140 +13)/2=153/2=76.5, which is above 76. So, beyond 76.5, profit would decrease. But the problem says that when x >76, profit decreases. So, for m=13, profit would start decreasing after 76.5, which is beyond 76. So, the condition is that the vertex must be at x ≤76, so that beyond 76, profit decreases.Hence, m must be ≤12.So, m is in (0,12].**Summary of Solutions:**1. The linear function is y = -2x + 200.2. The selling price that maximizes profit is 70 yuan per kg, with a maximum profit of 1800 yuan.3. The range of m is 0 < m ≤12.I think that's all. Let me just recap to ensure I didn't miss anything.For part 1, using two points to find the linear equation was straightforward.For part 2, setting up the profit function and finding its maximum via vertex was the key. Calculating the vertex correctly gave the optimal price and maximum profit.For part 3, adjusting the cost price by m and modifying the profit function accordingly, then ensuring that the vertex is at x ≤76 to satisfy the condition that profit decreases beyond 76 yuan. Solving for m gave the range.Everything seems consistent. I don't see any errors in my calculations or reasoning.

answer：First, I need to understand what it means to find the coordinates of the point ( P(-2, 1) ) with respect to the ( x )-axis.Reflecting a point across the ( x )-axis involves changing the sign of the ( y )-coordinate while keeping the ( x )-coordinate the same.So, for the point ( P(-2, 1) ):- The ( x )-coordinate remains ( -2 ).- The ( y )-coordinate becomes its opposite, so ( 1 ) becomes ( -1 ).Therefore, the coordinates of the point ( P ) with respect to the ( x )-axis are ( (-2, -1) ).Looking at the options provided:- A: ( (2, 1) )- B: ( (1, -2) )- C: ( (-2, -1) )- D: ( (2, -1) )The correct answer is option C.

answer：Okay, so I need to compute the definite integral from 1 to 3 of (2x minus 1 over x squared) dx. Hmm, let me think about how to approach this. I remember that to solve a definite integral, I first need to find the antiderivative of the function and then evaluate it at the upper and lower limits.Alright, let's break down the integrand into two separate parts: 2x and -1/x². I think I can find the antiderivative of each term separately and then combine them.Starting with the first term, 2x. The antiderivative of x with respect to x is (1/2)x², right? So if I have 2x, the antiderivative should be 2 times (1/2)x², which simplifies to x². That seems straightforward.Now, moving on to the second term, -1/x². Hmm, I need to find the antiderivative of -1/x². I recall that 1/x² is the same as x^(-2). The antiderivative of x^n is (x^(n+1))/(n+1), as long as n is not -1. So, applying that here, the antiderivative of x^(-2) would be (x^(-1))/(-1) which simplifies to -1/x. But since there's a negative sign in front of 1/x² in the original integrand, it becomes positive 1/x. So, the antiderivative of -1/x² is 1/x.Putting it all together, the antiderivative of the entire function 2x - 1/x² is x² + 1/x. I should probably add a constant of integration, but since we're dealing with a definite integral, the constants will cancel out when I evaluate at the limits, so I can ignore it for now.Now, I need to evaluate this antiderivative from 1 to 3. That means I'll plug in 3 into the antiderivative and then subtract the value of the antiderivative at 1.Let's calculate the antiderivative at the upper limit, which is 3. So, substituting x = 3 into x² + 1/x gives me 3² + 1/3. 3 squared is 9, and 1/3 is approximately 0.333... So, 9 + 0.333... equals 9.333..., which is the same as 28/3 if I convert it to an improper fraction. Wait, no, 9 is 27/3, so 27/3 + 1/3 is 28/3. Okay, that seems right.Next, I'll evaluate the antiderivative at the lower limit, which is 1. Substituting x = 1 into x² + 1/x gives me 1² + 1/1, which is 1 + 1, so that's 2. In fraction terms, 2 is 6/3.Now, I subtract the lower limit value from the upper limit value: 28/3 - 6/3. That equals 22/3. So, the definite integral from 1 to 3 of (2x - 1/x²) dx is 22/3.Let me just double-check my steps to make sure I didn't make any mistakes. First, I found the antiderivatives correctly: x² for 2x and 1/x for -1/x². Then, I evaluated them at 3 and 1, getting 28/3 and 6/3 respectively. Subtracting those gives 22/3, which seems right.I think I did everything correctly. The answer should be 22/3, which is option A.