💡Okay, so I have this problem where I need to find all functions ( f: mathbb{Z} to mathbb{Z} ) such that if ( x + y + z = 0 ), then ( f(x) + f(y) + f(z) = xyz ). Hmm, interesting. Let me try to figure this out step by step.First, I know that functional equations often require substituting specific values to find patterns or properties of the function. Maybe I can start by plugging in some simple values for ( x, y, z ) that satisfy ( x + y + z = 0 ).Let me try setting ( x = y = z = 0 ). Then, ( x + y + z = 0 ) is satisfied, and the equation becomes ( f(0) + f(0) + f(0) = 0 times 0 times 0 ), which simplifies to ( 3f(0) = 0 ). So, ( f(0) = 0 ). That's a start.Next, maybe I can set two variables equal and the third one to their negative. Let's say ( x = a ), ( y = a ), then ( z = -2a ) to satisfy ( x + y + z = 0 ). Plugging into the equation, we get ( f(a) + f(a) + f(-2a) = a times a times (-2a) ). Simplifying, that's ( 2f(a) + f(-2a) = -2a^3 ).Hmm, not sure if that helps directly, but maybe I can get more information by choosing other values. What if I set ( z = 0 )? Then ( x + y = 0 ), so ( y = -x ). Plugging into the equation, we get ( f(x) + f(-x) + f(0) = x times (-x) times 0 ). Since ( f(0) = 0 ), this simplifies to ( f(x) + f(-x) = 0 ). So, ( f ) is an odd function. That's a useful property.Knowing that ( f ) is odd, ( f(-x) = -f(x) ), which might help simplify things later.Now, let's think about the original condition again: ( x + y + z = 0 ) implies ( f(x) + f(y) + f(z) = xyz ). Maybe I can express this in terms of two variables instead of three. Let me set ( z = -x - y ), so the condition becomes ( f(x) + f(y) + f(-x - y) = x y (-x - y) ).Simplifying the right side: ( x y (-x - y) = -x^2 y - x y^2 ).So, ( f(x) + f(y) + f(-x - y) = -x^2 y - x y^2 ).But since ( f ) is odd, ( f(-x - y) = -f(x + y) ). So, substituting that in, we get:( f(x) + f(y) - f(x + y) = -x^2 y - x y^2 ).Rearranging, ( f(x + y) = f(x) + f(y) + x^2 y + x y^2 ).Hmm, this seems a bit complicated. Maybe I can try to find a form for ( f ). Let me assume that ( f ) is a polynomial function. Since the right side of the equation involves ( x^2 y ) and ( x y^2 ), which are degree 3 terms, perhaps ( f ) is a cubic polynomial.Let me suppose ( f(x) = a x^3 + b x^2 + c x + d ). But wait, since ( f ) is from ( mathbb{Z} ) to ( mathbb{Z} ), and we know ( f(0) = 0 ), so ( d = 0 ). Also, since ( f ) is odd, all the even-powered coefficients must be zero. So, ( b = 0 ). Therefore, ( f(x) = a x^3 + c x ).Let me test this form. Suppose ( f(x) = a x^3 + c x ). Then, let's plug into the equation ( f(x) + f(y) + f(z) = xyz ) when ( x + y + z = 0 ).So, ( f(x) + f(y) + f(z) = a(x^3 + y^3 + z^3) + c(x + y + z) ).But since ( x + y + z = 0 ), the second term becomes zero. So, we have ( a(x^3 + y^3 + z^3) = xyz ).I recall that there's an identity for ( x^3 + y^3 + z^3 ) when ( x + y + z = 0 ). Specifically, ( x^3 + y^3 + z^3 = 3xyz ). Let me verify that.Yes, indeed, if ( x + y + z = 0 ), then ( x^3 + y^3 + z^3 = 3xyz ). So, substituting back, we have ( a(3xyz) = xyz ). Therefore, ( 3a xyz = xyz ). For this to hold for all ( x, y, z ) with ( x + y + z = 0 ), we must have ( 3a = 1 ), so ( a = frac{1}{3} ).But wait, ( a ) must be an integer because ( f: mathbb{Z} to mathbb{Z} ). However, ( frac{1}{3} ) is not an integer. Hmm, that's a problem. Did I make a wrong assumption?Maybe assuming ( f ) is a cubic polynomial is too restrictive. Or perhaps I need to adjust my approach.Let me go back to the equation ( f(x + y) = f(x) + f(y) + x^2 y + x y^2 ). This looks like a modified Cauchy equation. Maybe I can find a function ( g(x) ) such that ( g(x + y) = g(x) + g(y) ), which is the standard Cauchy equation.Let me define ( g(x) = f(x) + k x^3 ), where ( k ) is a constant to be determined. Then, substituting into the equation:( g(x + y) = f(x + y) + k (x + y)^3 ).But from earlier, ( f(x + y) = f(x) + f(y) + x^2 y + x y^2 ). So,( g(x + y) = f(x) + f(y) + x^2 y + x y^2 + k (x^3 + 3x^2 y + 3x y^2 + y^3) ).Simplify:( g(x + y) = [f(x) + k x^3] + [f(y) + k y^3] + x^2 y + x y^2 + 3k x^2 y + 3k x y^2 ).Which is:( g(x + y) = g(x) + g(y) + x^2 y + x y^2 + 3k x^2 y + 3k x y^2 ).I want ( g(x + y) = g(x) + g(y) ), so the extra terms must cancel out. That is,( x^2 y + x y^2 + 3k x^2 y + 3k x y^2 = 0 ).Factor out ( x y ):( x y (x + y) + 3k x y (x + y) = 0 ).Factor out ( x y (x + y) ):( x y (x + y) (1 + 3k) = 0 ).Since this must hold for all ( x, y ), the coefficient must be zero. Therefore,( 1 + 3k = 0 implies k = -frac{1}{3} ).So, ( g(x) = f(x) - frac{1}{3} x^3 ). Then, ( g(x + y) = g(x) + g(y) ), which is Cauchy's functional equation.Since ( g ) is defined on integers, the solutions to Cauchy's equation are linear functions, i.e., ( g(x) = c x ) for some constant ( c in mathbb{Z} ).Therefore, ( f(x) = g(x) + frac{1}{3} x^3 = c x + frac{1}{3} x^3 ).But wait, ( f(x) ) must map integers to integers. So, ( frac{1}{3} x^3 ) must be an integer for all integer ( x ). Is that true?Let me check for some integer ( x ):- If ( x = 1 ), ( frac{1}{3} times 1 = frac{1}{3} ), which is not an integer.- If ( x = 2 ), ( frac{8}{3} ), also not an integer.- If ( x = 3 ), ( frac{27}{3} = 9 ), which is integer.Hmm, so it's only integer when ( x ) is a multiple of 3. But ( x ) can be any integer, so this suggests that ( frac{1}{3} x^3 ) isn't necessarily an integer. Therefore, my assumption that ( f(x) = c x + frac{1}{3} x^3 ) might not hold unless ( c ) compensates for the fractional part.Wait, but ( f(x) ) must be integer for all integer ( x ). So, ( frac{1}{3} x^3 ) must be integer for all ( x in mathbb{Z} ). But as I saw earlier, that's not the case. So, perhaps my approach is missing something.Maybe I need to adjust the function ( g(x) ) differently. Let me think again.Alternatively, perhaps ( f(x) ) has the form ( f(x) = frac{x^3 - x}{3} + c x ), where ( c ) is an integer. Let me test this.Compute ( f(x) = frac{x^3 - x}{3} + c x ).First, check if this is integer for all integer ( x ):( frac{x^3 - x}{3} = frac{x(x^2 - 1)}{3} = frac{x(x - 1)(x + 1)}{3} ).Since among three consecutive integers, one must be divisible by 3, so ( x(x - 1)(x + 1) ) is divisible by 3, making ( frac{x(x - 1)(x + 1)}{3} ) an integer. Therefore, ( f(x) ) is indeed an integer for all integer ( x ).Now, let's verify if this function satisfies the original condition.Given ( x + y + z = 0 ), we need to check if ( f(x) + f(y) + f(z) = xyz ).Compute ( f(x) + f(y) + f(z) ):( frac{x^3 - x}{3} + c x + frac{y^3 - y}{3} + c y + frac{z^3 - z}{3} + c z ).Combine terms:( frac{x^3 + y^3 + z^3 - (x + y + z)}{3} + c(x + y + z) ).But ( x + y + z = 0 ), so this simplifies to:( frac{x^3 + y^3 + z^3}{3} + 0 ).We know that ( x^3 + y^3 + z^3 = 3xyz ) when ( x + y + z = 0 ). Therefore,( frac{3xyz}{3} = xyz ).Which matches the required condition. So, this function works.Therefore, the general solution is ( f(x) = frac{x^3 - x}{3} + c x ) where ( c ) is an integer constant.I think that's it. So, all such functions are of the form ( f(x) = frac{x^3 - x}{3} + c x ) for some integer ( c ).

💡Okay, so I have this problem about two ships and a lighthouse. Let me try to visualize it first. There's a lighthouse that's 100 meters high, and two ships are on opposite sides of it. From each ship, the angle of elevation to the top of the lighthouse is measured. One of these angles is 30 degrees, and the other is unknown. The distance between the two ships is given as approximately 273.205 meters. I need to find the angle of elevation from the second ship.Alright, let's break this down. I think I can model this situation using right triangles. Each ship, the base of the lighthouse, and the top of the lighthouse form a right triangle. The height of the lighthouse is the opposite side for both angles of elevation, and the distances from each ship to the base of the lighthouse are the adjacent sides.So, for the first ship with the 30-degree angle, I can use the tangent function. Tangent of an angle in a right triangle is opposite over adjacent. That means tan(30°) = 100 / d1, where d1 is the distance from the first ship to the lighthouse.I remember that tan(30°) is equal to 1/√3, which is approximately 0.57735. So, if I rearrange the equation, d1 = 100 / tan(30°). Plugging in the value, d1 = 100 / (1/√3) = 100 * √3. Calculating that, √3 is approximately 1.732, so d1 ≈ 100 * 1.732 ≈ 173.2 meters.Now, the distance between the two ships is given as approximately 273.205 meters. If I subtract the distance from the first ship to the lighthouse from this total distance, I should get the distance from the second ship to the lighthouse. So, d2 = 273.205 - 173.2 ≈ 100 meters.Wait, that seems too clean. Is it exactly 100 meters? Let me check. If d1 is 100√3, which is approximately 173.205 meters, and the total distance between the ships is 273.205 meters, then d2 = 273.205 - 173.205 = 100 meters exactly. Okay, so the second ship is exactly 100 meters away from the lighthouse.Now, to find the angle of elevation from the second ship, I'll use the tangent function again. tan(theta2) = opposite / adjacent = 100 / 100 = 1. So, tan(theta2) = 1. I know that tan(45°) = 1, so theta2 must be 45 degrees.Hmm, that seems straightforward, but let me double-check. If the second ship is 100 meters away and the lighthouse is 100 meters tall, then the triangle formed is a 45-45-90 triangle, which makes sense. Both legs are equal, so the angles opposite them should be equal, hence 45 degrees each.But wait, is there another way to approach this problem? Maybe using the Law of Cosines or something else? Let me think. Since the two ships and the lighthouse form a triangle, with sides d1, d2, and the distance between the ships, which is 273.205 meters. I could use the Law of Cosines to find the angle at the lighthouse, but I'm not sure if that's necessary here because we already have the angles of elevation.Alternatively, I could consider the two right triangles separately and see if there's a relationship between them. But since we've already found the distances and one of the angles, it seems like using the tangent function was the most straightforward approach.I think I'm confident with the answer. The angle of elevation from the second ship is 45 degrees.

💡First, I need to calculate the squares of 103 and 97.Calculating (103^2):[103^2 = 103 times 103 = 10609]Calculating (97^2):[97^2 = 97 times 97 = 9409]Next, I find the difference between these squares:[10609 - 9409 = 1200]Since the result is already positive, the absolute value is:[|1200| = 1200]

💡First, I need to determine how many team members showed up to the game. Since each member scored 2 points and the total points scored were 6, I can divide the total points by the points per member to find the number of members who attended.Next, I'll subtract the number of members who showed up from the total number of team members to find out how many didn't attend the game.

💡Okay, so I have this problem where triangle ABC is shifted, and now the coordinates are given: A is at (2,8), B is at (2,2), and C is at (10,2). D and E are midpoints of AB and BC respectively. I need to find the point F where lines AE and CD intersect, and then find the sum of the x and y coordinates of F.First, I think I need to find the coordinates of points D and E since they are midpoints. To find the midpoint, I can use the midpoint formula, which is the average of the x-coordinates and the average of the y-coordinates of the endpoints.So, for point D, which is the midpoint of AB. Point A is (2,8) and point B is (2,2). Let me calculate the midpoint D:D_x = (2 + 2)/2 = 4/2 = 2D_y = (8 + 2)/2 = 10/2 = 5So, point D is at (2,5).Next, point E is the midpoint of BC. Point B is (2,2) and point C is (10,2). Let me calculate E:E_x = (2 + 10)/2 = 12/2 = 6E_y = (2 + 2)/2 = 4/2 = 2So, point E is at (6,2).Now, I need to find the equations of lines AE and CD because their intersection is point F.Let's start with line AE. Points A and E are (2,8) and (6,2) respectively. To find the equation of the line, I can use the slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept.First, let's find the slope (m) of line AE.Slope formula: m = (y2 - y1)/(x2 - x1)So, m = (2 - 8)/(6 - 2) = (-6)/4 = -3/2So, the slope of AE is -3/2.Now, using point A (2,8) to find the y-intercept (b):y = mx + b8 = (-3/2)(2) + b8 = -3 + bAdding 3 to both sides: 8 + 3 = b => b = 11So, the equation of line AE is y = (-3/2)x + 11.Now, let's find the equation of line CD. Points C and D are (10,2) and (2,5) respectively.Again, using the slope formula:m = (5 - 2)/(2 - 10) = 3/(-8) = -3/8Wait, that doesn't seem right. Let me double-check:Wait, point C is (10,2) and point D is (2,5). So, y2 - y1 is 5 - 2 = 3, and x2 - x1 is 2 - 10 = -8. So, slope m = 3/(-8) = -3/8. Hmm, that seems correct.Wait, but earlier, in the initial solution, the slope was calculated as -3/4. Did I make a mistake here?Wait, let me check again. Maybe I mixed up the points.Wait, point C is (10,2) and point D is (2,5). So, when calculating the slope, it's (y2 - y1)/(x2 - x1). So, if I take point D as (2,5) and point C as (10,2), then:m = (5 - 2)/(2 - 10) = 3/(-8) = -3/8.But in the initial solution, the slope was -3/4. Hmm, that suggests I might have made a mistake in the initial calculation.Wait, perhaps I confused points D and C? Let me see.Wait, point D is the midpoint of AB, which is (2,5), and point C is (10,2). So, line CD connects (2,5) and (10,2). So, the slope is (2 - 5)/(10 - 2) = (-3)/8 = -3/8.Wait, so in the initial solution, the slope was calculated as -3/4, which is incorrect. So, that might have led to an error in the intersection point.Wait, so I need to correct that.So, the slope of CD is -3/8, not -3/4.Wait, let me recalculate the equation of line CD with the correct slope.So, using point C (10,2) and slope -3/8.Using point-slope form: y - y1 = m(x - x1)So, y - 2 = (-3/8)(x - 10)Simplify:y = (-3/8)x + (30/8) + 2Convert 2 to eighths: 2 = 16/8So, y = (-3/8)x + 30/8 + 16/8 = (-3/8)x + 46/8 = (-3/8)x + 23/4So, the equation of line CD is y = (-3/8)x + 23/4.Wait, but in the initial solution, the equation was given as y = (-3/4)x + 17/2, which is different.So, perhaps the initial solution had an error in calculating the slope.So, moving forward, I need to correct that.So, now, I have the equations of lines AE and CD.Line AE: y = (-3/2)x + 11Line CD: y = (-3/8)x + 23/4Now, to find the intersection point F, I need to set these two equations equal to each other and solve for x.So:(-3/2)x + 11 = (-3/8)x + 23/4Let me solve for x.First, let's get rid of the fractions by multiplying both sides by 8 to eliminate denominators.8*(-3/2)x + 8*11 = 8*(-3/8)x + 8*(23/4)Simplify each term:8*(-3/2)x = -12x8*11 = 888*(-3/8)x = -3x8*(23/4) = 2*23 = 46So, the equation becomes:-12x + 88 = -3x + 46Now, let's bring all terms to one side.-12x + 88 + 3x - 46 = 0Combine like terms:(-12x + 3x) + (88 - 46) = 0-9x + 42 = 0Now, solve for x:-9x = -42x = (-42)/(-9) = 42/9 = 14/3 ≈ 4.666...So, x = 14/3.Now, substitute x back into one of the equations to find y. Let's use line AE: y = (-3/2)x + 11.So, y = (-3/2)*(14/3) + 11Simplify:(-3/2)*(14/3) = (-14/2) = -7So, y = -7 + 11 = 4Therefore, point F is at (14/3, 4).Now, the problem asks for the sum of the x and y coordinates of F.So, 14/3 + 4 = 14/3 + 12/3 = 26/3 ≈ 8.666...But let me check my calculations again because the initial solution had a different result, and I want to make sure I didn't make a mistake.Wait, let me verify the equations again.Line AE: from (2,8) to (6,2). Slope is (2-8)/(6-2) = (-6)/4 = -3/2. Correct.Equation: y = (-3/2)x + b. Plugging in (2,8):8 = (-3/2)*2 + b => 8 = -3 + b => b = 11. So, equation is y = (-3/2)x + 11. Correct.Line CD: from (10,2) to (2,5). Slope is (5-2)/(2-10) = 3/(-8) = -3/8. Correct.Equation: y = (-3/8)x + b. Plugging in (10,2):2 = (-3/8)*10 + b => 2 = -30/8 + b => 2 = -15/4 + b => b = 2 + 15/4 = 8/4 + 15/4 = 23/4. So, equation is y = (-3/8)x + 23/4. Correct.Setting them equal:(-3/2)x + 11 = (-3/8)x + 23/4Multiply both sides by 8:-12x + 88 = -3x + 46-12x + 88 = -3x + 46Bring variables to left and constants to right:-12x + 3x = 46 - 88-9x = -42x = (-42)/(-9) = 14/3. Correct.Then y = (-3/2)*(14/3) + 11 = (-14/2) + 11 = -7 + 11 = 4. Correct.So, point F is at (14/3, 4). The sum is 14/3 + 4 = 14/3 + 12/3 = 26/3.Wait, but in the initial solution, the point F was at (-4,17), which seems way off. So, perhaps the initial solution had a mistake in calculating the slope of CD.Wait, in the initial solution, the slope of CD was calculated as (2 - 5)/(6 - 2) = (-3)/4 = -3/4, which is incorrect because point D is (2,5) and point C is (10,2), so the change in x is 2 - 10 = -8, not 6 - 2 = 4. So, that was a mistake in the initial solution.Therefore, the correct intersection point is (14/3, 4), and the sum is 26/3.But wait, 26/3 is approximately 8.666..., which is a fractional value. The problem didn't specify whether the answer needs to be an integer or not, so 26/3 is acceptable.Wait, but let me check if I made any other mistakes.Wait, when I calculated the slope of CD, I used points C (10,2) and D (2,5). So, the change in y is 5 - 2 = 3, and change in x is 2 - 10 = -8, so slope is 3/(-8) = -3/8. Correct.Equation of CD: y = (-3/8)x + 23/4. Correct.Equation of AE: y = (-3/2)x + 11. Correct.Intersection: solved for x = 14/3, y = 4. Correct.So, the sum is 14/3 + 4 = 26/3.Wait, but 26/3 is about 8.666..., which is not an integer. The initial solution had -4 + 17 = 13, which is an integer. So, perhaps I made a mistake in the calculation.Wait, let me check the equations again.Line AE: from (2,8) to (6,2). Slope is (2-8)/(6-2) = -6/4 = -3/2. Correct.Equation: y = (-3/2)x + b. Plugging in (2,8):8 = (-3/2)*2 + b => 8 = -3 + b => b = 11. Correct.Line CD: from (10,2) to (2,5). Slope is (5-2)/(2-10) = 3/(-8) = -3/8. Correct.Equation: y = (-3/8)x + b. Plugging in (10,2):2 = (-3/8)*10 + b => 2 = -30/8 + b => 2 = -15/4 + b => b = 2 + 15/4 = 8/4 + 15/4 = 23/4. Correct.So, equations are correct.Setting them equal:(-3/2)x + 11 = (-3/8)x + 23/4Multiply both sides by 8:-12x + 88 = -3x + 46-12x + 88 = -3x + 46Bring variables to left:-12x + 3x = 46 - 88-9x = -42x = (-42)/(-9) = 14/3. Correct.Then y = (-3/2)*(14/3) + 11 = (-14/2) + 11 = -7 + 11 = 4. Correct.So, point F is at (14/3, 4). The sum is 14/3 + 4 = 26/3.Wait, but 26/3 is approximately 8.666..., which is a fractional value. The initial solution had an integer result, but that was due to an error in calculating the slope of CD.Therefore, the correct answer is 26/3, which is approximately 8.666...But let me check if I can express 26/3 as a mixed number: 8 and 2/3.Alternatively, as an improper fraction, it's 26/3.But the problem asks for the sum of the x and y coordinates, so it's 14/3 + 4 = 26/3.Alternatively, if I made a mistake in the initial setup, perhaps the lines AE and CD are not correctly defined.Wait, let me confirm the points again.Point A is (2,8), point E is the midpoint of BC, which is (6,2). So, line AE connects (2,8) to (6,2). Correct.Point C is (10,2), point D is the midpoint of AB, which is (2,5). So, line CD connects (10,2) to (2,5). Correct.So, the equations are correct.Therefore, the intersection point F is at (14/3, 4), and the sum is 26/3.Wait, but 26/3 is approximately 8.666..., which is not an integer. The initial solution had an integer result, but that was due to an error in the slope calculation.Therefore, the correct answer is 26/3.But let me check if I can express this as a mixed number or if it's better to leave it as an improper fraction.26 divided by 3 is 8 with a remainder of 2, so 8 and 2/3. But since the problem doesn't specify, either form is acceptable, but in mathematical problems, improper fractions are often preferred.Therefore, the sum is 26/3.Wait, but I just realized that in the initial solution, the point F was at (-4,17), which is way outside the triangle, while in my corrected solution, it's at (14/3,4), which is inside the triangle. That makes more sense because the intersection of medians should be inside the triangle.Wait, but in this case, AE and CD are not medians, because D is the midpoint of AB and E is the midpoint of BC, so lines AE and CD are actually medians.Wait, no, in a triangle, a median connects a vertex to the midpoint of the opposite side. So, AE connects A to E, the midpoint of BC, so that's a median. Similarly, CD connects C to D, the midpoint of AB, so that's also a median.Therefore, the intersection point F is the centroid of the triangle, which divides each median into a ratio of 2:1.Wait, but in my corrected solution, the centroid is at (14/3,4). Let me check if that's correct.The centroid of a triangle is the average of the coordinates of the vertices.So, centroid (G) is at:G_x = (2 + 2 + 10)/3 = 14/3 ≈ 4.666...G_y = (8 + 2 + 2)/3 = 12/3 = 4.So, centroid is at (14/3,4), which matches my corrected solution.Therefore, point F is indeed the centroid, and its coordinates are (14/3,4), so the sum is 14/3 + 4 = 26/3.Therefore, the correct answer is 26/3.But wait, the initial solution had an answer of 13, which is 26/2, but that was due to an error in the slope calculation.Therefore, the correct sum is 26/3.But let me confirm once more.Given that the centroid is at (14/3,4), the sum is 14/3 + 4 = 14/3 + 12/3 = 26/3.Yes, that's correct.So, the final answer is 26/3, which can be written as boxed{dfrac{26}{3}}.But wait, the problem says "the sum of the x and y coordinates of F". So, 14/3 + 4 = 26/3, which is approximately 8.666...But perhaps I should write it as a fraction.Yes, 26/3 is the exact value.Therefore, the sum is 26/3.**Final Answer**The sum of the x and y coordinates of point ( F ) is boxed{dfrac{26}{3}}.

💡Okay, so I have this problem here about the function ( y = |log_2 x| ). The domain is given as ([a, b]) and the range is ([0, 2]). I need to find the difference between the maximum and minimum lengths of the interval ([a, b]). Hmm, let me try to break this down step by step.First, I remember that the length of an interval ([x_1, x_2]) is just (x_2 - x_1). So, I need to figure out what possible intervals ([a, b]) can make the function ( y = |log_2 x| ) have a range of ([0, 2]). Then, among all those possible intervals, I need to find the one with the maximum length and the one with the minimum length, and subtract them to get the difference.Alright, let's start by understanding the function ( y = |log_2 x| ). The logarithm function ( log_2 x ) is defined for ( x > 0 ). It's negative when ( 0 < x < 1 ) and positive when ( x > 1 ). Taking the absolute value of it, ( |log_2 x| ), will make all the negative values positive. So, the graph of ( y = |log_2 x| ) will be V-shaped, with the vertex at ( x = 1 ), where ( y = 0 ).Now, the range of the function is given as ([0, 2]). That means the smallest value ( y ) can take is 0, and the largest is 2. So, I need to find all ( x ) such that ( |log_2 x| ) is between 0 and 2.Let me solve the inequality ( 0 leq |log_2 x| leq 2 ). This can be split into two cases because of the absolute value.Case 1: ( log_2 x geq 0 ). Then, ( |log_2 x| = log_2 x ). So, the inequality becomes ( 0 leq log_2 x leq 2 ). Converting this to exponential form, we get ( 2^0 leq x leq 2^2 ), which simplifies to ( 1 leq x leq 4 ).Case 2: ( log_2 x < 0 ). Then, ( |log_2 x| = -log_2 x ). So, the inequality becomes ( 0 leq -log_2 x leq 2 ). Multiplying all parts by -1 (and remembering to reverse the inequality signs), we get ( 0 geq log_2 x geq -2 ). Converting this to exponential form, ( 2^{-2} leq x leq 2^0 ), which simplifies to ( frac{1}{4} leq x leq 1 ).So, combining both cases, the domain of ( x ) that satisfies ( 0 leq |log_2 x| leq 2 ) is ( frac{1}{4} leq x leq 4 ). That is, the interval ([ frac{1}{4}, 4 ]).But wait, the problem says the domain is ([a, b]). So, does that mean ([a, b]) has to be exactly ([ frac{1}{4}, 4 ])? Or can it be any interval within that?I think it can be any interval within ([ frac{1}{4}, 4 ]) as long as the range of ( y ) is still ([0, 2]). So, to get the range ([0, 2]), the domain ([a, b]) must include all the necessary ( x ) values that make ( y ) reach 0 and 2.Let me think. The minimum value of ( y ) is 0, which occurs at ( x = 1 ). The maximum value of ( y ) is 2, which occurs at ( x = frac{1}{4} ) and ( x = 4 ). So, to have the range ([0, 2]), the domain ([a, b]) must include 1 (to get 0) and at least one of ( frac{1}{4} ) or 4 (to get 2). But wait, actually, to cover the entire range from 0 to 2, the domain must include all the points where ( y ) can vary between 0 and 2.So, if I take an interval that starts at ( frac{1}{4} ) and goes up to 4, that's the full interval where ( y ) goes from 2 down to 0 and back up to 2. But if I take a smaller interval, say from ( frac{1}{4} ) to 1, then ( y ) will go from 2 down to 0. Similarly, from 1 to 4, ( y ) will go from 0 up to 2. So, both intervals ([ frac{1}{4}, 1 ]) and ([1, 4]) will have the range ([0, 2]).Therefore, the possible intervals ([a, b]) can be either ([ frac{1}{4}, 1 ]), ([1, 4]), or the entire interval ([ frac{1}{4}, 4 ]). But wait, could there be other intervals that also result in the range ([0, 2])?Hmm, let's see. Suppose I take an interval that doesn't include 1. For example, ([ frac{1}{4}, 2 ]). Then, ( y ) would go from 2 down to 0 at ( x = 1 ) and then back up to ( log_2 2 = 1 ). So, the range would be ([0, 2]), because at ( x = frac{1}{4} ), ( y = 2 ), and at ( x = 1 ), ( y = 0 ). So, even though the interval doesn't go all the way to 4, it still covers the necessary points to have the range ([0, 2]).Wait, so actually, any interval that includes 1 and either ( frac{1}{4} ) or 4, or both, will have the range ([0, 2]). So, the minimal intervals would be those that just include 1 and either ( frac{1}{4} ) or 4, but not necessarily both.But hold on, if I take an interval that starts somewhere between ( frac{1}{4} ) and 1 and ends somewhere between 1 and 4, will that still cover the range ([0, 2])?Let me test with an example. Suppose I take ([ frac{1}{2}, 2 ]). Then, ( y ) at ( x = frac{1}{2} ) is ( |log_2 frac{1}{2}| = 1 ), at ( x = 1 ) is 0, and at ( x = 2 ) is 1. So, the range here is ([0, 1]), which is less than ([0, 2]). So, that interval doesn't work.So, in order to have the range ([0, 2]), the interval must include either ( frac{1}{4} ) or 4, because those are the points where ( y = 2 ). If the interval doesn't include either ( frac{1}{4} ) or 4, then the maximum ( y ) would be less than 2.Therefore, the interval ([a, b]) must include 1 (to get ( y = 0 )) and at least one of ( frac{1}{4} ) or 4 (to get ( y = 2 )). So, the minimal intervals would be those that just include 1 and either ( frac{1}{4} ) or 4. The maximum interval would be the entire ([ frac{1}{4}, 4 ]).So, let's figure out the lengths.First, the maximum length is when the interval is ([ frac{1}{4}, 4 ]). The length is ( 4 - frac{1}{4} = frac{16}{4} - frac{1}{4} = frac{15}{4} ).Now, for the minimum length. The minimal intervals would be those that just include 1 and either ( frac{1}{4} ) or 4. So, the interval ([ frac{1}{4}, 1 ]) has length ( 1 - frac{1}{4} = frac{3}{4} ), and the interval ([1, 4]) has length ( 4 - 1 = 3 ).Wait, so the minimal length is ( frac{3}{4} ), and the maximum is ( frac{15}{4} ). Therefore, the difference between the maximum and minimum lengths is ( frac{15}{4} - frac{3}{4} = frac{12}{4} = 3 ).But hold on, is there a possibility of a smaller interval? For example, if I take an interval that includes 1 and a point slightly beyond ( frac{1}{4} ) or 4, but not the entire way. But wait, if I take an interval that doesn't include ( frac{1}{4} ) or 4, then the maximum ( y ) would be less than 2, so the range wouldn't be ([0, 2]). Therefore, the interval must include at least one of those points.So, the minimal interval is indeed ([ frac{1}{4}, 1 ]) with length ( frac{3}{4} ), and the maximum interval is ([ frac{1}{4}, 4 ]) with length ( frac{15}{4} ). The difference is 3.Wait, but earlier I thought the interval ([1, 4]) has length 3, which is larger than ( frac{3}{4} ). So, actually, the minimal interval is ( frac{3}{4} ), and the maximum is ( frac{15}{4} ). So, their difference is 3.But let me double-check. If I take the interval ([ frac{1}{4}, 1 ]), the range is ([0, 2]), right? Because at ( x = frac{1}{4} ), ( y = 2 ), and at ( x = 1 ), ( y = 0 ). So, yes, that works.Similarly, the interval ([1, 4]) also gives the range ([0, 2]). So, both of these intervals have lengths ( frac{3}{4} ) and 3, respectively. Wait, but 3 is larger than ( frac{3}{4} ). So, the maximum length is ( frac{15}{4} ), which is 3.75, and the minimum is ( frac{3}{4} ), which is 0.75. So, the difference is ( 3.75 - 0.75 = 3 ).Yes, that makes sense. So, the difference between the maximum and minimum lengths is 3.I think that's the answer. Let me just recap:- The function ( y = |log_2 x| ) has a range of ([0, 2]) when ( x ) is in ([ frac{1}{4}, 4 ]).- The maximum interval is ([ frac{1}{4}, 4 ]) with length ( frac{15}{4} ).- The minimum interval is either ([ frac{1}{4}, 1 ]) or ([1, 4]), both with lengths ( frac{3}{4} ) and 3, respectively. Wait, no, actually, the minimal interval is ([ frac{1}{4}, 1 ]) with length ( frac{3}{4} ), and the maximum is ([ frac{1}{4}, 4 ]) with length ( frac{15}{4} ).- The difference is ( frac{15}{4} - frac{3}{4} = 3 ).Yes, that seems correct.