💡Okay, so I have this problem here about functions defined on all real numbers, and I need to figure out which one has the smallest positive period of π. The options are A: y = sin|x|, B: y = cos|x|, C: y = |sinx|, and D: y = |cos2x|. Hmm, let's break this down step by step.First, I remember that the period of a function is the smallest positive number T such that the function repeats every T units. For example, the basic sine and cosine functions have a period of 2π. But when you take the absolute value of these functions or modify them in other ways, their periods can change. So, I need to analyze each option one by one.Starting with option A: y = sin|x|. Wait, the absolute value inside the sine function. I think that affects the graph. Normally, sinx is an odd function, meaning sin(-x) = -sinx. But when you take sin|x|, it becomes an even function because |x| is always positive, so sin|x| = sinx for x ≥ 0 and sin(-x) = sinx for x < 0. So, the graph of sin|x| would be symmetric about the y-axis. But does that affect the period? Let me think. The period of sinx is 2π, but since we're reflecting the negative side to the positive side, does it make the function repeat more often? Hmm, actually, I don't think so. Because sin|x| is just the right half of the sine wave mirrored on the left side. So, it doesn't really repeat every π or anything like that. It still has the same period as sinx, which is 2π. So, A probably doesn't have a period of π.Moving on to option B: y = cos|x|. Similar idea here. Cosine is already an even function, so cos|x| is just cosx. Because cosine is even, cos(-x) = cosx, so cos|x| = cosx for all x. Therefore, the period of cos|x| is the same as the period of cosx, which is 2π. So, B also doesn't have a period of π.Option C: y = |sinx|. Okay, here we're taking the absolute value of sinx. Let me recall what that does. The sine function normally oscillates between -1 and 1. Taking the absolute value would reflect the negative parts to positive, effectively making the function non-negative. So, the graph of |sinx| would look like a series of "V" shapes, peaking at 1 and going back down to 0. Now, how does this affect the period? Well, the original sinx has a period of 2π, but when we take the absolute value, the function repeats its pattern every π instead of 2π. Because the negative half of the sine wave is flipped to be positive, so the pattern from 0 to π is the same as from π to 2π, but mirrored. Therefore, the period is halved. So, the smallest positive period for |sinx| is π. That seems to fit the condition given in the problem. So, option C is a candidate.Option D: y = |cos2x|. Let's analyze this. First, the cosine function with a coefficient inside. The general form is cos(kx), which has a period of 2π/k. So, for cos2x, the period would be 2π/2 = π. Now, taking the absolute value of cos2x. Similar to the previous case, the absolute value would reflect the negative parts of the cosine wave to positive. So, the function |cos2x| would have a period that is half of the original period of cos2x. Since cos2x has a period of π, |cos2x| would have a period of π/2. Wait, is that right? Let me think. If the original function repeats every π, then taking absolute value would make it repeat every π/2 because the negative half is now positive, so the pattern repeats twice as fast. So, yes, the period becomes π/2. Therefore, the smallest positive period for |cos2x| is π/2, which is smaller than π. So, D doesn't satisfy the condition of having the smallest positive period of π.So, summarizing:- A: Period is 2π- B: Period is 2π- C: Period is π- D: Period is π/2Therefore, the function that satisfies the condition is option C: y = |sinx|.Wait, just to make sure I didn't make a mistake with option D. Let me double-check. The function is |cos2x|. The period of cos2x is π, as 2π divided by 2 is π. Taking the absolute value would cause the function to repeat its pattern every half period, so π/2. That seems correct. So, yes, D has a smaller period than π.And for option C, |sinx|, the original period is 2π, but taking absolute value makes it π. That seems right because the negative part is flipped, so the function repeats every π instead of 2π.So, I think I'm confident that option C is the correct answer.**Final Answer**boxed{C}

💡Okay, so I've got this probability problem here, and I need to find z(x|y) and z(x|y) again. Hmm, that seems a bit repetitive, but maybe it's a typo or something. Let me see what information I've been given.First, I know that z(x) = 0.02. I'm assuming z(x) represents the probability of event x occurring, so P(x) = 0.02. Similarly, z(y) = 0.10, so P(y) = 0.10. Then, z(xny) = 0.10. I think "xny" might be a typo or shorthand for the intersection of x and y, so P(x ∩ y) = 0.10. Wait a minute, if P(x ∩ y) is 0.10, but P(x) is only 0.02, that doesn't seem right. How can the probability of both x and y happening be higher than the probability of x alone? That doesn't make sense because the intersection of two events can't have a higher probability than either of the individual events. Maybe there's a mistake in the given values?Let me double-check. If P(x) = 0.02 and P(y) = 0.10, then P(x ∩ y) should be less than or equal to both P(x) and P(y). But here, P(x ∩ y) is given as 0.10, which is greater than P(x). That's impossible because the intersection can't have a higher probability than either event individually. There must be an error in the problem statement.Assuming that P(x ∩ y) is actually supposed to be 0.02, which would make sense because it can't exceed P(x), let's proceed with that. So, P(x ∩ y) = 0.02.Now, I need to find P(x|y), which is the probability of x given y. The formula for conditional probability is:P(x|y) = P(x ∩ y) / P(y)Plugging in the numbers:P(x|y) = 0.02 / 0.10 = 0.2So, P(x|y) is 0.2.But the problem asks for z(x|y) twice. Maybe there was a mistake in the problem statement, and it was supposed to ask for both P(x|y) and P(y|x)? Let's check P(y|x) as well.P(y|x) = P(x ∩ y) / P(x)Using the same values:P(y|x) = 0.02 / 0.02 = 1So, P(y|x) is 1, which means that if x occurs, y is certain to occur.But going back to the original problem, since it specifically asks for z(x|y) twice, I think there might have been an intention to ask for both P(x|y) and P(y|x). Given that, I'll provide both results.In summary:- P(x|y) = 0.2- P(y|x) = 1This makes sense because if P(y|x) is 1, it means that whenever x happens, y always happens. And since P(x ∩ y) is 0.02, which is the same as P(x), it confirms that x is a subset of y in this scenario.I should also note that the initial values provided seem inconsistent because P(x ∩ y) was given as 0.10, which is higher than P(x). This inconsistency makes the problem impossible as stated. Therefore, correcting P(x ∩ y) to 0.02 aligns the probabilities logically.So, with the corrected values:- P(x|y) = 0.2- P(y|x) = 1This provides a coherent solution to the problem.

💡Alright, so I have this problem where I need to find the expected value of the determinant of ( A - A^t ), where ( A ) is a ( 2n times 2n ) matrix with entries chosen independently at random to be either 0 or 1, each with probability 1/2. Hmm, okay. Let me try to unpack this step by step.First, let me understand what ( A - A^t ) is. ( A^t ) is the transpose of ( A ), so the entry in the ( i )-th row and ( j )-th column of ( A^t ) is the same as the entry in the ( j )-th row and ( i )-th column of ( A ). Therefore, the matrix ( X = A - A^t ) will have entries ( x_{i,j} = a_{i,j} - a_{j,i} ). Wait, so each entry ( x_{i,j} ) is the difference between the corresponding entry in ( A ) and its transpose. That means ( x_{i,j} = -x_{j,i} ) because ( x_{j,i} = a_{j,i} - a_{i,j} = -(a_{i,j} - a_{j,i}) = -x_{i,j} ). So, ( X ) is a skew-symmetric matrix. I remember that skew-symmetric matrices have some special properties. For example, all their eigenvalues are either zero or purely imaginary, and the determinant of a skew-symmetric matrix of odd order is zero. But in this case, ( X ) is a ( 2n times 2n ) matrix, which is even-dimensional, so its determinant might not necessarily be zero. Interesting.So, the problem reduces to finding the expected value of the determinant of a random skew-symmetric matrix ( X ) where each entry ( x_{i,j} ) for ( i < j ) is ( a_{i,j} - a_{j,i} ), and each ( a_{i,j} ) is 0 or 1 with equal probability. Since ( x_{i,j} ) and ( x_{j,i} ) are related, they are not independent. Let me think about the entries of ( X ). For ( i neq j ), ( x_{i,j} ) can be either -1, 0, or 1. Specifically, ( x_{i,j} = 1 ) if ( a_{i,j} = 1 ) and ( a_{j,i} = 0 ), ( x_{i,j} = -1 ) if ( a_{i,j} = 0 ) and ( a_{j,i} = 1 ), and ( x_{i,j} = 0 ) if both ( a_{i,j} ) and ( a_{j,i} ) are equal (either both 0 or both 1). So, each ( x_{i,j} ) has a 1/2 chance of being 0, a 1/4 chance of being 1, and a 1/4 chance of being -1. That's because the entries ( a_{i,j} ) and ( a_{j,i} ) are independent, each with probability 1/2 of being 0 or 1. So, the probability that ( a_{i,j} = 1 ) and ( a_{j,i} = 0 ) is 1/4, and similarly for the other case.But wait, actually, since ( x_{i,j} = a_{i,j} - a_{j,i} ), the possible values are -1, 0, or 1. Let me compute the probabilities:- ( x_{i,j} = 1 ) if ( a_{i,j} = 1 ) and ( a_{j,i} = 0 ): probability ( frac{1}{2} times frac{1}{2} = frac{1}{4} ).- ( x_{i,j} = -1 ) if ( a_{i,j} = 0 ) and ( a_{j,i} = 1 ): probability ( frac{1}{2} times frac{1}{2} = frac{1}{4} ).- ( x_{i,j} = 0 ) otherwise: probability ( 1 - frac{1}{4} - frac{1}{4} = frac{1}{2} ).So, each off-diagonal entry ( x_{i,j} ) is 1 with probability 1/4, -1 with probability 1/4, and 0 with probability 1/2. The diagonal entries ( x_{i,i} ) are always 0 because ( x_{i,i} = a_{i,i} - a_{i,i} = 0 ).Now, I need to compute the expected value of ( det(X) ). The determinant of a matrix is a sum over all permutations, involving products of entries and signs based on the permutation's inversion count. Specifically, for a matrix ( X ), the determinant is:[det(X) = sum_{sigma in S_{2n}} text{sgn}(sigma) prod_{i=1}^{2n} x_{i, sigma(i)}]where ( S_{2n} ) is the symmetric group of all permutations of ( 2n ) elements, and ( text{sgn}(sigma) ) is the sign of the permutation ( sigma ), which is ( (-1)^{text{number of inversions in } sigma} ).So, the expected value ( mathbb{E}[det(X)] ) is the sum over all permutations ( sigma ) of ( text{sgn}(sigma) ) times the expectation of the product ( prod_{i=1}^{2n} x_{i, sigma(i)} ).Mathematically, this is:[mathbb{E}[det(X)] = sum_{sigma in S_{2n}} text{sgn}(sigma) mathbb{E}left[ prod_{i=1}^{2n} x_{i, sigma(i)} right]]Now, the expectation of the product is the product of expectations if the variables are independent. However, in this case, the entries ( x_{i,j} ) are not all independent because ( x_{i,j} = -x_{j,i} ). So, the entries are dependent in pairs.Therefore, for the expectation ( mathbb{E}left[ prod_{i=1}^{2n} x_{i, sigma(i)} right] ) to be non-zero, the permutation ( sigma ) must pair up the indices such that each transposition is accounted for. Specifically, each ( x_{i, sigma(i)} ) must be paired with its transpose ( x_{sigma(i), i} ), which is equal to ( -x_{i, sigma(i)} ).Wait, so if ( sigma ) is a permutation where each element is either a fixed point or part of a transposition, then the product ( prod_{i=1}^{2n} x_{i, sigma(i)} ) will involve pairs like ( x_{i,j} x_{j,i} ). But since ( x_{j,i} = -x_{i,j} ), each such pair becomes ( x_{i,j} (-x_{i,j}) = -x_{i,j}^2 ).But ( x_{i,j} ) can be -1, 0, or 1. So, ( x_{i,j}^2 ) is 1 if ( x_{i,j} ) is 1 or -1, and 0 otherwise. Therefore, ( x_{i,j} x_{j,i} = -x_{i,j}^2 ) is either -1 or 0.So, for the expectation ( mathbb{E}left[ prod_{i=1}^{2n} x_{i, sigma(i)} right] ) to be non-zero, the permutation ( sigma ) must consist only of fixed points and transpositions. However, since ( X ) is skew-symmetric, the diagonal entries are zero, so any fixed point in the permutation would result in a factor of zero in the product. Therefore, the permutation ( sigma ) must consist solely of transpositions, i.e., it must be a product of disjoint transpositions with no fixed points.Such permutations are called "fixed-point-free involutions" or "perfect matchings." The number of such permutations in ( S_{2n} ) is ( (2n - 1)!! ), where ( !! ) denotes the double factorial. The double factorial ( (2n - 1)!! ) is the product of all odd integers up to ( 2n - 1 ).So, for each such permutation ( sigma ), the product ( prod_{i=1}^{2n} x_{i, sigma(i)} ) is equal to ( (-1)^n prod_{i=1}^{n} x_{i,j}^2 ), since each transposition contributes a factor of ( -x_{i,j}^2 ). Therefore, the expectation of this product is ( (-1)^n prod_{i=1}^{n} mathbb{E}[x_{i,j}^2] ).But ( x_{i,j}^2 ) is 1 with probability ( frac{1}{2} ) (since ( x_{i,j} ) is 1 or -1 each with probability 1/4, so total 1/2) and 0 with probability 1/2. Therefore, ( mathbb{E}[x_{i,j}^2] = frac{1}{2} times 1 + frac{1}{2} times 0 = frac{1}{2} ).Thus, the expectation of the product is ( (-1)^n left( frac{1}{2} right)^n ).Putting it all together, the expected determinant is:[mathbb{E}[det(X)] = sum_{sigma in text{fixed-point-free involutions}} text{sgn}(sigma) times (-1)^n left( frac{1}{2} right)^n]Now, I need to figure out the sign of each permutation ( sigma ). A permutation consisting of ( n ) disjoint transpositions has a sign of ( (-1)^n ), since each transposition contributes a factor of -1, and there are ( n ) transpositions. Therefore, ( text{sgn}(sigma) = (-1)^n ).So, each term in the sum is ( (-1)^n times (-1)^n times left( frac{1}{2} right)^n = left( frac{1}{2} right)^n ).The number of such permutations ( sigma ) is ( (2n - 1)!! ). Therefore, the expected value is:[mathbb{E}[det(X)] = (2n - 1)!! times left( frac{1}{2} right)^n]Simplifying, this is:[mathbb{E}[det(A - A^t)] = frac{(2n - 1)!!}{2^n}]So, that's the expected value of the determinant of ( A - A^t ) as a function of ( n ).**Final Answer**The expected value of ( det(A - A^t) ) is boxed{dfrac{(2n-1)!!}{2^n}}.

💡Okay, so I have this function f(x) = 3x - x³, and I need to do two things with it. First, find the equation of the tangent line at the point (2, f(2)), and second, figure out where the function is increasing or decreasing. Hmm, let's start with the first part.Alright, to find the tangent line at a specific point, I remember that I need two things: the slope of the tangent line at that point and a point through which the tangent line passes. They've given me the point (2, f(2)), so I just need to calculate f(2) to get the y-coordinate. Let me do that.f(2) = 3*(2) - (2)³ = 6 - 8 = -2. So the point is (2, -2). Got that down.Now, for the slope of the tangent line at x=2. I think I need to find the derivative of f(x) because the derivative at a point gives the slope of the tangent line there. Let me compute f'(x).f(x) = 3x - x³, so f'(x) is the derivative of 3x minus the derivative of x³. The derivative of 3x is 3, and the derivative of x³ is 3x². So f'(x) = 3 - 3x². That seems right.Now, plug in x=2 into the derivative to get the slope at that point. So f'(2) = 3 - 3*(2)² = 3 - 3*4 = 3 - 12 = -9. So the slope of the tangent line at x=2 is -9.Alright, now I have the slope and a point on the line. I can use the point-slope form of a line to write the equation. The point-slope formula is y - y1 = m(x - x1), where m is the slope and (x1, y1) is the point.Plugging in the values, we get y - (-2) = -9(x - 2). Simplifying that, y + 2 = -9x + 18. Then subtract 2 from both sides: y = -9x + 16. Wait, is that right? Let me check my math again.Wait, y - (-2) is y + 2, and -9(x - 2) is -9x + 18. So y + 2 = -9x + 18. Subtracting 2 from both sides gives y = -9x + 16. Hmm, but when I plug x=2 into this equation, y should be -2. Let's check: y = -9*(2) + 16 = -18 + 16 = -2. Okay, that matches. So the equation of the tangent line is y = -9x + 16.Wait, but I thought earlier I had y = -9x + 18. Did I make a mistake somewhere? Let me go back. When I calculated f(2), I got -2, that's correct. The derivative at 2 was -9, that's right. Then, using point-slope: y - (-2) = -9(x - 2). So y + 2 = -9x + 18. Then y = -9x + 18 - 2, which is y = -9x + 16. Yeah, so I think I made a mistake in my initial thought where I thought it was y = -9x + 18. It's actually y = -9x + 16. Okay, so that's the tangent line.Alright, moving on to part two: determining where the function is increasing or decreasing. I remember that if the derivative is positive, the function is increasing, and if the derivative is negative, the function is decreasing. So I need to analyze the sign of f'(x).We already found f'(x) = 3 - 3x². Let me factor that to make it easier to find critical points. So f'(x) = 3(1 - x²) = 3(1 - x)(1 + x). So the critical points are where f'(x) = 0, which happens when 1 - x = 0 or 1 + x = 0. So x = 1 and x = -1.These critical points divide the real number line into intervals. I need to test each interval to see if the derivative is positive or negative there. The intervals are:1. (-∞, -1)2. (-1, 1)3. (1, ∞)Let me pick test points in each interval.For (-∞, -1), let's choose x = -2. Plugging into f'(x): 3 - 3*(-2)² = 3 - 3*4 = 3 - 12 = -9, which is negative. So the function is decreasing on (-∞, -1).For (-1, 1), let's choose x = 0. Plugging into f'(x): 3 - 3*(0)² = 3 - 0 = 3, which is positive. So the function is increasing on (-1, 1).For (1, ∞), let's choose x = 2. Plugging into f'(x): 3 - 3*(2)² = 3 - 12 = -9, which is negative. So the function is decreasing on (1, ∞).So putting it all together, f(x) is decreasing on (-∞, -1), increasing on (-1, 1), and decreasing again on (1, ∞).Wait, let me double-check. At x = -2, f'(x) is negative, so decreasing. At x=0, positive, increasing. At x=2, negative, decreasing. Yep, that seems consistent.So, to summarize:1. The tangent line at (2, f(2)) is y = -9x + 16.2. The function f(x) is decreasing on (-∞, -1), increasing on (-1, 1), and decreasing on (1, ∞).I think that's it. I don't see any mistakes in my calculations now.

💡Okay, so I have this problem here about functions and their inverses. Let me try to understand it step by step. The problem says:"Let ( y = f(x) ) have an inverse function ( y = f^{-1}(x) ). Also, ( y = f(x + 2) ) and ( y = f^{-1}(x - 1) ) are inverse functions of each other. Find the value of ( f^{-1}(2004) - f^{-1}(1) )."And the options are:A: 4006 B: 4008 C: 2003 D: 2004Alright, so first, I know that if ( y = f(x) ) has an inverse ( y = f^{-1}(x) ), then by definition, ( f(f^{-1}(x)) = x ) and ( f^{-1}(f(x)) = x ). That's the basic property of inverse functions.Now, the problem introduces two new functions: ( y = f(x + 2) ) and ( y = f^{-1}(x - 1) ), and says they are inverses of each other. So, that means if I compose them, I should get the identity function. Let me write that out:If ( y = f(x + 2) ) and ( y = f^{-1}(x - 1) ) are inverses, then:1. ( f(f^{-1}(x - 1) + 2) = x )2. ( f^{-1}(f(x + 2) - 1) = x )Hmm, maybe I can use one of these equations to find a relationship between ( f ) and ( f^{-1} ).Let me take the first equation: ( f(f^{-1}(x - 1) + 2) = x ). Let me simplify the inside first. Let me denote ( z = f^{-1}(x - 1) ). Then, ( f(z + 2) = x ). But since ( z = f^{-1}(x - 1) ), then ( f(z) = x - 1 ). So, ( f(z + 2) = x ).So, ( f(z + 2) = x ), but ( f(z) = x - 1 ). Therefore, ( f(z + 2) = f(z) + 1 ). So, ( f(z + 2) = f(z) + 1 ).This seems like a functional equation. It suggests that adding 2 to the input of ( f ) increases the output by 1. So, ( f ) is a function with a constant rate of change—maybe linear?Wait, if ( f(z + 2) = f(z) + 1 ), then the function ( f ) has a slope of ( frac{1}{2} ). Because for every increase of 2 in ( z ), ( f(z) ) increases by 1. So, if ( f ) is linear, it would be of the form ( f(z) = frac{1}{2}z + c ), where ( c ) is a constant.Let me test this assumption. Suppose ( f(z) = frac{1}{2}z + c ). Then, ( f^{-1}(x) ) would be ( 2(x - c) ), since to find the inverse, we solve for ( z ) in ( x = frac{1}{2}z + c ), which gives ( z = 2(x - c) ).Now, let's check if ( y = f(x + 2) ) and ( y = f^{-1}(x - 1) ) are inverses. First, compute ( f(x + 2) ):( f(x + 2) = frac{1}{2}(x + 2) + c = frac{1}{2}x + 1 + c ).Then, ( f^{-1}(x - 1) = 2((x - 1) - c) = 2x - 2 - 2c ).Now, let's see if they are inverses. Let me compose ( f(x + 2) ) and ( f^{-1}(x - 1) ):First, apply ( f^{-1}(x - 1) ) to some input ( a ):( f^{-1}(a - 1) = 2(a - 1) - 2c = 2a - 2 - 2c ).Then, apply ( f(x + 2) ) to this result:( f( (2a - 2 - 2c) + 2 ) = f(2a - 2c) = frac{1}{2}(2a - 2c) + c = a - c + c = a ).So, that works out. Similarly, if I compose the other way around:Apply ( f(x + 2) ) to some input ( b ):( f(b + 2) = frac{1}{2}(b + 2) + c = frac{1}{2}b + 1 + c ).Then, apply ( f^{-1}(x - 1) ) to this result:( f^{-1}( (frac{1}{2}b + 1 + c) - 1 ) = f^{-1}(frac{1}{2}b + c) = 2( frac{1}{2}b + c - c ) = 2( frac{1}{2}b ) = b ).So, that also works. Therefore, my assumption that ( f ) is linear with slope ( frac{1}{2} ) seems correct.Now, knowing that ( f(z) = frac{1}{2}z + c ), let's find ( f^{-1}(x) ). As I found earlier, ( f^{-1}(x) = 2(x - c) ).But I need to find ( f^{-1}(2004) - f^{-1}(1) ). Let's compute that:( f^{-1}(2004) = 2(2004 - c) )( f^{-1}(1) = 2(1 - c) )So, subtracting them:( f^{-1}(2004) - f^{-1}(1) = 2(2004 - c) - 2(1 - c) = 2*2004 - 2c - 2 + 2c = 4008 - 2 = 4006 ).Wait, that's 4006, which is option A.But let me double-check. Maybe I made a mistake in assuming ( f ) is linear. The problem didn't specify that ( f ) is linear, only that it has an inverse. So, maybe I need a different approach.Let me go back to the original condition. We have ( y = f(x + 2) ) and ( y = f^{-1}(x - 1) ) are inverses. So, by definition, ( f(f^{-1}(x - 1) + 2) = x ) and ( f^{-1}(f(x + 2) - 1) = x ).Let me focus on the first equation: ( f(f^{-1}(x - 1) + 2) = x ).Let me denote ( u = f^{-1}(x - 1) ). Then, ( f(u + 2) = x ). But since ( u = f^{-1}(x - 1) ), then ( f(u) = x - 1 ). So, ( f(u + 2) = x ).Therefore, ( f(u + 2) = f(u) + 1 ). So, ( f(u + 2) - f(u) = 1 ).This suggests that the function ( f ) increases by 1 when its argument increases by 2. So, the function has a constant difference over a constant interval, which is characteristic of a linear function with slope ( frac{1}{2} ). So, even without assuming linearity, this functional equation suggests that ( f ) is linear with slope ( frac{1}{2} ).Therefore, my earlier conclusion that ( f ) is linear with slope ( frac{1}{2} ) seems valid.Thus, ( f^{-1}(x) = 2(x - c) ), and the difference ( f^{-1}(2004) - f^{-1}(1) = 4006 ).Wait, but let me think again. The problem didn't specify any particular value for ( c ), so how come the difference doesn't depend on ( c )? Because when I subtract ( f^{-1}(2004) - f^{-1}(1) ), the constants cancel out:( 2(2004 - c) - 2(1 - c) = 4008 - 2c - 2 + 2c = 4006 ).Yes, the ( c ) terms cancel, so the difference is indeed 4006, regardless of the value of ( c ).Therefore, the answer should be 4006, which is option A.But just to be thorough, let me consider another approach without assuming linearity.Suppose I let ( g(x) = f(x + 2) ) and ( h(x) = f^{-1}(x - 1) ). Since ( g ) and ( h ) are inverses, ( g(h(x)) = x ) and ( h(g(x)) = x ).Let me compute ( g(h(x)) ):( g(h(x)) = f(h(x) + 2) = f(f^{-1}(x - 1) + 2) ).Let me set ( y = f^{-1}(x - 1) ), so ( f(y) = x - 1 ). Then, ( f(y + 2) = x ).Therefore, ( f(y + 2) = f(y) + 1 ), as before.So, this again leads to the conclusion that ( f(y + 2) - f(y) = 1 ), which suggests a linear function with slope ( frac{1}{2} ).Thus, regardless of the approach, the conclusion is the same.Therefore, I'm confident that the answer is 4006, which is option A.

💡Okay, so I have this geometry problem here with three distinct lines m, n, and l, and a plane α. The problem states that l is perpendicular to m, and I need to figure out which of the four given statements (A, B, C, D) is correct. Let me try to break this down step by step.First, I'll try to visualize the scenario. We have three lines and a plane. Line l is perpendicular to line m. That means they intersect at a right angle, right? So, if I imagine line m lying on a table, line l would be like a vertical pole sticking up from the table where it meets m.Now, let's look at each option one by one.**Option A: If m is not contained in α and l is perpendicular to α, then m is parallel to α.**Hmm, okay. So, m is not in the plane α, and l is perpendicular to the entire plane α. If l is perpendicular to α, that means l is perpendicular to every line in α that it intersects. Since l is also perpendicular to m, does that mean m is parallel to α?Wait, if l is perpendicular to α, then l is like a normal vector to the plane. If m is not in α and l is perpendicular to m, then m can't intersect α because if it did, l would have to be perpendicular to that intersection point as well, which might not necessarily be the case. So, maybe m is parallel to α? That seems plausible. I'll keep this in mind.**Option B: If l is perpendicular to n, then m is perpendicular to n.**Alright, so if l is perpendicular to n, does that mean m is also perpendicular to n? Well, l is perpendicular to m, and l is perpendicular to n. Does that imply m is perpendicular to n?I'm not sure. In three-dimensional space, if two lines are both perpendicular to a third line, they don't necessarily have to be perpendicular to each other. They could be skew lines or just any other orientation. So, I don't think this is necessarily true. Maybe they could be parallel or something else. So, I think B might not be correct.**Option C: If l is perpendicular to n, then m is parallel to n.**Okay, similar to B, but now it's saying m is parallel to n. So, if l is perpendicular to both m and n, does that make m and n parallel?Again, in three dimensions, if two lines are both perpendicular to a third line, they don't have to be parallel. They could intersect or be skew. So, I don't think this is necessarily true either. So, C might not be correct.**Option D: If m is parallel to n and n is contained in α, then l is perpendicular to α.**Alright, so m is parallel to n, and n is in plane α. Does that mean l is perpendicular to α?Well, l is perpendicular to m. If m is parallel to n, then l should also be perpendicular to n, right? Because if two lines are parallel, any line perpendicular to one should be perpendicular to the other.But does that mean l is perpendicular to the entire plane α? For l to be perpendicular to α, it needs to be perpendicular to every line in α, not just n. Since n is just one line in α, l being perpendicular to n doesn't necessarily mean it's perpendicular to all lines in α. So, I don't think D is correct.Going back to Option A, I think that might be the right answer. If l is perpendicular to α, then l is a normal vector to the plane. Since l is also perpendicular to m, and m is not in α, then m can't intersect α because if it did, l would have to be perpendicular to that intersection, which might not hold. Therefore, m must be parallel to α.Wait, but is that always true? Let me think. If m is not in α and l is perpendicular to both m and α, then m must be parallel to α. Yeah, that makes sense because if m weren't parallel, it would intersect α at some point, and l would have to be perpendicular to that intersection, which would complicate things. So, yeah, I think A is correct.So, after going through each option, I believe the correct answer is A.