💡Okay, so I have this problem about a parallelepiped with volume V, and I need to find the volume of a specific pyramid inside it, which is pyramid ABCC₁. Hmm, first, I should probably recall what a parallelepiped is. I think it's like a 3D figure formed by six parallelograms. So, it's kind of like a cube but with all sides being parallelograms instead of squares.Alright, the volume of the parallelepiped is given as V. Now, I need to find the volume of pyramid ABCC₁. Let me visualize this. Pyramid ABCC₁ has a base that's a triangle, right? Because it's connecting points A, B, C, and then up to C₁. Wait, actually, no. Let me think again. Pyramid ABCC₁ is formed by the points A, B, C, and C₁. So, the base is triangle ABC, and the apex is C₁.Wait, is that correct? Or is the base a quadrilateral? Hmm, no, because it's connecting A, B, C, and C₁, so it's a four-faced figure. So, the base is triangle ABC, and the apex is C₁. So, it's a tetrahedron with base ABC and apex C₁.Okay, so to find the volume of this pyramid, I need to remember the formula for the volume of a pyramid. I think it's one-third times the area of the base times the height. So, Volume = (1/3) * Base Area * Height.Now, the base is triangle ABC. Since ABCD is a face of the parallelepiped, which is a parallelogram, triangle ABC would have half the area of the parallelogram ABCD. So, if the area of ABCD is S, then the area of ABC is S/2.But wait, I don't know the exact area of ABCD. However, I do know the volume of the parallelepiped, which is V. The volume of a parallelepiped is given by the base area times the height. So, V = S * h, where S is the area of the base and h is the height perpendicular to the base.So, if I can express the volume of the pyramid in terms of V, that would solve the problem. Let's see. The volume of the pyramid is (1/3) * (S/2) * h. Because the base area is S/2 and the height is h, which is the same as the height of the parallelepiped.So, substituting, Volume = (1/3) * (S/2) * h = (1/6) * S * h. But since S * h is equal to V, the volume of the parallelepiped, then the volume of the pyramid is (1/6) * V.Wait, does that make sense? Let me double-check. If I have a parallelepiped, and I take a pyramid that's a portion of it, the volume should be a fraction of V. Since the pyramid is occupying a portion of the space, and considering the base is half of the parallelogram and the height is the same, the factor of 1/6 seems reasonable.Alternatively, I can think of the parallelepiped being divided into six such pyramids. If I consider all possible pyramids formed by connecting each vertex to the opposite vertex, there would be six of them, each with the same volume. So, each would have a volume of V/6.Yeah, that seems to align with my earlier calculation. So, I think the volume of pyramid ABCC₁ is V/6.**Final Answer**The volume of the pyramid (ABCC_1) is boxed{dfrac{V}{6}}.

💡First, I need to determine how many ways the parents can assign different service providers to each of their four children from the 25 available providers.For Laura, the first child, there are 25 possible service providers to choose from.Once a provider is chosen for Laura, there are 24 remaining providers for the second child.After selecting providers for the first two children, there are 23 providers left for the third child.Finally, for the fourth child, there are 22 providers remaining.To find the total number of ways to assign the providers, I multiply the number of choices for each child:25 (for Laura) × 24 (for the second child) × 23 (for the third child) × 22 (for the fourth child) = 303,600.Therefore, there are 303,600 different ways the parents can choose the service providers for their four children.

💡Okay, so I have this problem where I need to find g(2021) and g(1/2021) given that g is a function from [0, ∞) to [0, ∞) satisfying the functional equation g(g(x)) = 3x/(x + 3) for all x ≥ 0. Additionally, I know that for 2 ≤ x ≤ 3, g(x) = (x + 1)/2. Hmm, functional equations can be tricky, but maybe I can figure out a pattern or find a way to express g(x) in terms of x. Let me start by understanding the given functional equation better. The equation is g(g(x)) = 3x/(x + 3). So, if I apply g twice to any x, I get 3x/(x + 3). That seems like a kind of recursive relationship. Maybe I can find an expression for g(x) by considering some substitution or transformation.I also know that for x between 2 and 3, g(x) is (x + 1)/2. That might be useful. Let me see if I can extend this definition beyond the interval [2, 3]. Maybe I can use the functional equation to find g(x) for x outside this interval.Let me try plugging in x = 2 into the functional equation. If x = 2, then g(g(2)) = 3*2/(2 + 3) = 6/5 = 1.2. But since 2 is in the interval [2, 3], I know that g(2) = (2 + 1)/2 = 1.5. So, g(1.5) should be 1.2. Wait, so g(1.5) = 1.2. That gives me another point. Maybe I can find a pattern here. Let me try to compute a few more points. If I take x = 1.5, then g(g(1.5)) = 3*1.5/(1.5 + 3) = 4.5/4.5 = 1. So, g(1.2) = 1. Hmm, so g(1.2) = 1. Then, if I plug x = 1.2 into the functional equation, g(g(1.2)) = 3*1.2/(1.2 + 3) = 3.6/4.2 = 6/7 ≈ 0.857. But g(1.2) = 1, so g(1) = 6/7 ≈ 0.857.Wait, so g(1) = 6/7. Then, plugging x = 1 into the functional equation, g(g(1)) = 3*1/(1 + 3) = 3/4 = 0.75. So, g(6/7) = 0.75.Continuing this way, g(6/7) = 0.75. Then, plugging x = 6/7 into the functional equation, g(g(6/7)) = 3*(6/7)/(6/7 + 3) = (18/7)/(6/7 + 21/7) = (18/7)/(27/7) = 18/27 = 2/3 ≈ 0.6667. So, g(0.75) = 2/3.This seems like it's creating a sequence of values. Maybe I can see a pattern here. Let me list out the values I have so far:- g(2) = 1.5- g(1.5) = 1.2- g(1.2) = 1- g(1) = 6/7 ≈ 0.857- g(6/7) = 0.75- g(0.75) = 2/3 ≈ 0.6667It looks like each time, g is taking a value and mapping it to a smaller value, approaching some limit. Maybe this is part of a cycle or a converging sequence. But I need to find g(2021) and g(1/2021). 2021 is a large number, and 1/2021 is very small. So, I need to figure out how g behaves for large x and for small x.Let me think about the functional equation again: g(g(x)) = 3x/(x + 3). If I let x be very large, say x approaches infinity, then 3x/(x + 3) approaches 3. So, g(g(x)) approaches 3 as x becomes large. Similarly, if x is very small, approaching 0, then 3x/(x + 3) approaches 0. So, g(g(x)) approaches 0 as x approaches 0.Hmm, so for very large x, g(g(x)) is approaching 3, which suggests that g(x) might be approaching some value that maps to 3 under g. Similarly, for very small x, g(g(x)) approaches 0, so g(x) might be approaching some value that maps to 0 under g.Wait, maybe I can find an expression for g(x) in terms of x. Let me assume that g(x) is of the form kx/(x + a) for some constants k and a. Let me test this assumption.Suppose g(x) = kx/(x + a). Then, g(g(x)) would be g(kx/(x + a)) = k*(kx/(x + a)) / (kx/(x + a) + a) = k^2 x / (kx + a(x + a)).Simplify the denominator: kx + a(x + a) = (k + a)x + a^2.So, g(g(x)) = k^2 x / [(k + a)x + a^2].We know that g(g(x)) = 3x/(x + 3). So, setting these equal:k^2 x / [(k + a)x + a^2] = 3x/(x + 3).Cancel x from both sides (assuming x ≠ 0):k^2 / [(k + a)x + a^2] = 3/(x + 3).Cross-multiplying:k^2 (x + 3) = 3[(k + a)x + a^2].Expand both sides:k^2 x + 3k^2 = 3(k + a)x + 3a^2.Now, equate coefficients of x and the constants:For x: k^2 = 3(k + a).For constants: 3k^2 = 3a^2 ⇒ k^2 = a^2 ⇒ k = ±a.Since g maps [0, ∞) to [0, ∞), and given the functional equation, k and a must be positive. So, k = a.Substituting k = a into the equation for x:k^2 = 3(k + k) ⇒ k^2 = 6k ⇒ k^2 - 6k = 0 ⇒ k(k - 6) = 0.So, k = 0 or k = 6. But k = 0 would make g(x) = 0 for all x, which doesn't satisfy the functional equation because g(g(x)) would be 0, not 3x/(x + 3). So, k = 6.Thus, a = k = 6. Therefore, g(x) = 6x/(x + 6).Wait, does this hold? Let me check:g(g(x)) = g(6x/(x + 6)) = 6*(6x/(x + 6)) / (6x/(x + 6) + 6).Simplify numerator: 36x/(x + 6).Denominator: (6x + 6(x + 6))/(x + 6) = (6x + 6x + 36)/(x + 6) = (12x + 36)/(x + 6) = 12(x + 3)/(x + 6).So, g(g(x)) = (36x/(x + 6)) / (12(x + 3)/(x + 6)) = (36x)/(12(x + 3)) = 3x/(x + 3). Perfect, that matches the functional equation.So, it seems that g(x) = 6x/(x + 6) for all x ≥ 0. But wait, the problem states that for 2 ≤ x ≤ 3, g(x) = (x + 1)/2. Does this match with g(x) = 6x/(x + 6)?Let me check for x = 2: g(2) = (2 + 1)/2 = 1.5. Using g(x) = 6x/(x + 6): 6*2/(2 + 6) = 12/8 = 1.5. Okay, that matches.For x = 3: g(3) = (3 + 1)/2 = 2. Using g(x) = 6x/(x + 6): 6*3/(3 + 6) = 18/9 = 2. That also matches.So, actually, the function g(x) = 6x/(x + 6) satisfies both the functional equation and the given condition for 2 ≤ x ≤ 3. Therefore, g(x) = 6x/(x + 6) for all x ≥ 0.Wait, that simplifies things a lot! So, I don't need to do all that earlier step-by-step computation. I can just use this expression to find g(2021) and g(1/2021).Let me compute g(2021):g(2021) = 6*2021 / (2021 + 6) = 6*2021 / 2027.Similarly, g(1/2021) = 6*(1/2021) / (1/2021 + 6) = (6/2021) / (1/2021 + 6).Let me compute these.First, g(2021):6*2021 = 12126.2021 + 6 = 2027.So, g(2021) = 12126 / 2027.Let me divide 12126 by 2027:2027 * 6 = 12162, which is larger than 12126.So, 2027 * 5 = 10135.12126 - 10135 = 1991.Wait, that doesn't seem right. Maybe I should compute 12126 ÷ 2027.Let me see:2027 * 6 = 12162, which is 36 more than 12126.So, 12126 = 2027*6 - 36.Therefore, 12126 / 2027 = 6 - 36/2027.Simplify 36/2027: 36 ÷ 2027 ≈ 0.01776.So, g(2021) ≈ 6 - 0.01776 ≈ 5.98224.But maybe I can write it as a fraction:12126 / 2027 = (2027*6 - 36)/2027 = 6 - 36/2027.So, g(2021) = 6 - 36/2027.Alternatively, I can leave it as 12126/2027, but perhaps simplifying:Divide numerator and denominator by GCD(12126, 2027). Let's find GCD(2027, 12126).Using Euclidean algorithm:12126 ÷ 2027 = 5 with remainder 12126 - 5*2027 = 12126 - 10135 = 1991.Now, GCD(2027, 1991).2027 ÷ 1991 = 1 with remainder 36.GCD(1991, 36).1991 ÷ 36 = 55 with remainder 1 (since 55*36 = 1980, 1991 - 1980 = 11? Wait, 55*36=1980, 1991-1980=11. Hmm, wait, 1991 ÷ 36 is 55 with remainder 11.Wait, 36*55=1980, 1991-1980=11.So, GCD(36,11).36 ÷ 11 = 3 with remainder 3.GCD(11,3).11 ÷ 3 = 3 with remainder 2.GCD(3,2).3 ÷ 2 = 1 with remainder 1.GCD(2,1).GCD is 1.So, GCD(12126,2027)=1. Therefore, 12126/2027 is already in simplest terms.So, g(2021)=12126/2027.Similarly, for g(1/2021):g(1/2021)=6*(1/2021)/(1/2021 +6)= (6/2021)/(1/2021 +6).Let me compute the denominator:1/2021 +6 = (1 + 6*2021)/2021 = (1 + 12126)/2021 = 12127/2021.So, g(1/2021)= (6/2021)/(12127/2021)=6/12127.Simplify 6/12127. Let's see if it can be reduced.Find GCD(6,12127). 12127 ÷6=2021 with remainder 1 (since 6*2021=12126, 12127-12126=1). So, GCD is 1.Thus, g(1/2021)=6/12127.Wait, but earlier I thought g(x)=6x/(x+6). Let me check if this holds for x=1/2021.Yes, g(1/2021)=6*(1/2021)/(1/2021 +6)=6/(1 +6*2021)=6/12127.So, that's correct.But wait, earlier when I tried to compute g(1/2021) using the transformation h(x), I got a different result. Let me check that.Wait, in the initial problem, the assistant introduced a function h(x)=1/g(1/x). Then, through some steps, they found h(2021)=2021.5, which led to g(2021)=2021.5.But according to my current approach, g(x)=6x/(x+6), which gives g(2021)=12126/2027≈5.98224, which is approximately 6, but not exactly 2021.5.This discrepancy suggests that my assumption that g(x)=6x/(x+6) for all x might be incorrect, despite it satisfying the functional equation and the given interval.Wait, hold on. The problem states that g: [0, ∞) → [0, ∞) satisfies g(g(x))=3x/(x+3) for all x≥0, and for 2≤x≤3, g(x)=(x+1)/2.I assumed that g(x)=6x/(x+6) for all x, but perhaps that's not the case. Maybe g(x) is defined piecewise, with g(x)=6x/(x+6) outside the interval [2,3], and g(x)=(x+1)/2 within [2,3].Wait, but when I checked x=2 and x=3, both definitions give the same result. For x=2: (2+1)/2=1.5 and 6*2/(2+6)=12/8=1.5. Similarly, x=3: (3+1)/2=2 and 6*3/(3+6)=18/9=2. So, the function is continuous at x=2 and x=3.Therefore, perhaps g(x)=6x/(x+6) for all x≥0, including the interval [2,3]. But wait, the problem says "for 2≤x≤3, g(x)=(x+1)/2". So, is this an additional condition, or is it part of the definition?Wait, maybe the function is defined as g(x)=(x+1)/2 for 2≤x≤3, and elsewhere it's defined by the functional equation. So, perhaps g(x)=6x/(x+6) only holds outside [2,3], but within [2,3], it's (x+1)/2.But when I checked x=2 and x=3, both expressions give the same result, so it's consistent.Wait, but if g(x)=6x/(x+6) for all x, then for x in [2,3], it's equal to (x+1)/2. Let me check for x=2.5:g(2.5)=6*2.5/(2.5+6)=15/8.5≈1.7647.But (2.5 +1)/2=3.5/2=1.75.Wait, 15/8.5≈1.7647≠1.75.So, they are not equal. Therefore, my initial assumption that g(x)=6x/(x+6) for all x is incorrect. The function must be defined piecewise, with g(x)=(x+1)/2 for 2≤x≤3, and another expression outside this interval.So, I need to find the expression for g(x) outside [2,3]. Let me try to find it.Given that g(g(x))=3x/(x+3), and for x in [2,3], g(x)=(x+1)/2. Let me see if I can find g(x) for x outside [2,3].Let me consider x >3. Let me pick x=4. Then, g(g(4))=3*4/(4+3)=12/7≈1.714.But I don't know g(4). Let me denote y=g(4). Then, g(y)=12/7.But y=g(4). If I can find y such that g(y)=12/7.Now, 12/7≈1.714, which is less than 2, so y must be such that g(y)=12/7. Since y is in [0, ∞), and g maps [0, ∞) to [0, ∞), y could be in [0,2) or [2,3] or (3, ∞).But if y is in [2,3], then g(y)=(y+1)/2. So, if g(y)=12/7, then (y+1)/2=12/7 ⇒ y+1=24/7 ⇒ y=24/7 -1=17/7≈2.4286.So, y=17/7≈2.4286, which is in [2,3]. Therefore, g(4)=17/7.So, g(4)=17/7≈2.4286.Similarly, let me try x=5.g(g(5))=3*5/(5+3)=15/8=1.875.Let y=g(5). Then, g(y)=15/8=1.875.Since 1.875 <2, y must be such that g(y)=1.875.If y is in [2,3], then g(y)=(y+1)/2=1.875 ⇒ y+1=3.75 ⇒ y=2.75.So, y=2.75, which is in [2,3]. Therefore, g(5)=2.75.Wait, so for x=4, g(4)=17/7≈2.4286, and for x=5, g(5)=2.75.Hmm, seems like for x>3, g(x) is increasing but less than 3.Wait, let me see if I can find a pattern or a formula for g(x) when x>3.Suppose x>3, then g(x)=?From the above examples, for x=4, g(4)=17/7≈2.4286, which is (4 + something)/something.Wait, 17/7 is approximately 2.4286, which is (4 + 1)/ (something). Wait, 17/7= (4 + 1)/ (something). Hmm, not sure.Alternatively, maybe g(x) for x>3 is given by (3x)/(x + 3). Wait, let me test that.If g(x)=3x/(x + 3), then g(g(x))=g(3x/(x + 3))=3*(3x/(x + 3))/(3x/(x + 3) +3)=9x/(x + 3)/( (3x + 3(x + 3))/(x + 3))=9x/(x + 3)/( (6x +9)/(x + 3))=9x/(6x +9)=3x/(2x +3).But we need g(g(x))=3x/(x +3). So, 3x/(2x +3)≠3x/(x +3). Therefore, that doesn't work.Alternatively, maybe g(x) for x>3 is (x +1)/2. But for x=4, (4 +1)/2=2.5, but we found g(4)=17/7≈2.4286≠2.5. So, that's not it.Wait, perhaps g(x) for x>3 is defined such that g(g(x))=3x/(x +3). So, if I let g(x)=y, then g(y)=3x/(x +3). So, for x>3, y=g(x) must satisfy g(y)=3x/(x +3). Since x>3, 3x/(x +3) is less than 3, because 3x/(x +3)=3 - 9/(x +3). So, as x increases, 3x/(x +3) approaches 3.Therefore, y=g(x) must be such that g(y)=3x/(x +3). Since 3x/(x +3) is less than 3, y must be in [0,3). But for y in [2,3], g(y)=(y +1)/2. For y in [0,2), we don't have a definition yet.Wait, so perhaps for x>3, g(x) is in [2,3), because g(y)=3x/(x +3) is less than 3, and if y is in [2,3), then g(y)=(y +1)/2.Wait, let me formalize this.Suppose x>3. Let y=g(x). Then, g(y)=3x/(x +3). Since x>3, 3x/(x +3) is in (1.5, 3). Specifically, when x approaches 3 from above, 3x/(x +3) approaches 9/6=1.5, and as x approaches infinity, 3x/(x +3) approaches 3.Therefore, y must be in [2,3) because g(y)=3x/(x +3) is in (1.5,3), but for y in [2,3), g(y)=(y +1)/2, which is in [1.5,2). Wait, but 3x/(x +3) is in (1.5,3), so if g(y)=3x/(x +3), and y is in [2,3), then g(y)=(y +1)/2 must equal 3x/(x +3). Therefore, (y +1)/2=3x/(x +3).So, solving for y:(y +1)/2=3x/(x +3) ⇒ y +1=6x/(x +3) ⇒ y=6x/(x +3) -1= (6x - (x +3))/(x +3)= (5x -3)/(x +3).Therefore, y=(5x -3)/(x +3).But y=g(x). So, for x>3, g(x)=(5x -3)/(x +3).Let me check this with x=4:g(4)=(5*4 -3)/(4 +3)=(20 -3)/7=17/7≈2.4286, which matches our earlier result.Similarly, for x=5:g(5)=(5*5 -3)/(5 +3)=(25 -3)/8=22/8=11/4=2.75, which also matches.Good, so for x>3, g(x)=(5x -3)/(x +3).Now, what about x<2? Let me consider x=1.We know that g(g(1))=3*1/(1 +3)=3/4=0.75.Let y=g(1). Then, g(y)=0.75.Since y=g(1), and 1<2, y must be in [0, ∞). But for y in [2,3], g(y)=(y +1)/2. If y is in [0,2), we don't have a definition yet.Wait, but g(y)=0.75. If y is in [2,3], then g(y)=(y +1)/2=0.75 ⇒ y +1=1.5 ⇒ y=0.5. But 0.5 is not in [2,3], so that's a contradiction.Therefore, y must be in [0,2). So, for y in [0,2), we need to define g(y)=0.75.But wait, we don't have a definition for g(y) when y is in [0,2). So, perhaps we need to define g(x) for x in [0,2) such that g(g(x))=3x/(x +3).Let me denote for x in [0,2), g(x)=z, then g(z)=3x/(x +3). Since z is in [0, ∞), and if z is in [2,3], then g(z)=(z +1)/2. If z is in [0,2), we need to define g(z).Wait, this is getting complicated. Maybe I can find an expression for g(x) when x is in [0,2).Let me assume that for x in [0,2), g(x)=kx/(x + a), similar to before. Let me test this.Suppose g(x)=kx/(x + a) for x in [0,2). Then, g(g(x))=g(kx/(x + a))=k*(kx/(x + a))/(kx/(x + a) + a)=k^2 x / (kx + a(x + a)).We know that g(g(x))=3x/(x +3). So,k^2 x / (kx + a(x + a)) = 3x/(x +3).Cancel x:k^2 / (kx + a(x + a)) = 3/(x +3).Cross-multiplying:k^2 (x +3) = 3(kx + a(x + a)).Expand:k^2 x + 3k^2 = 3k x + 3a x + 3a^2.Group like terms:(k^2 - 3k - 3a)x + (3k^2 - 3a^2)=0.Since this must hold for all x in [0,2), the coefficients must be zero:1. k^2 - 3k - 3a =0.2. 3k^2 - 3a^2=0 ⇒ k^2 = a^2 ⇒ k=±a.Since g maps [0, ∞) to [0, ∞), k and a must be positive. So, k=a.Substitute k=a into the first equation:a^2 - 3a - 3a=0 ⇒ a^2 -6a=0 ⇒ a(a -6)=0.Thus, a=0 or a=6. But a=0 would make g(x)=0 for all x, which doesn't satisfy the functional equation. So, a=6, hence k=6.Therefore, for x in [0,2), g(x)=6x/(x +6).Wait, let me check this. For x=1, g(1)=6*1/(1 +6)=6/7≈0.857. Then, g(g(1))=g(6/7)=6*(6/7)/(6/7 +6)=36/7 / (6/7 +42/7)=36/7 / 48/7=36/48=3/4=0.75, which matches the functional equation.Similarly, for x=0.5, g(0.5)=6*0.5/(0.5 +6)=3/6.5≈0.4615. Then, g(g(0.5))=g(3/6.5)=6*(3/6.5)/(3/6.5 +6)=18/6.5 / (3/6.5 +39/6.5)=18/6.5 /42/6.5=18/42=3/7≈0.4286.But according to the functional equation, g(g(0.5))=3*0.5/(0.5 +3)=1.5/3.5≈0.4286. So, it matches.Therefore, for x in [0,2), g(x)=6x/(x +6).So, summarizing:- For x in [0,2), g(x)=6x/(x +6).- For x in [2,3], g(x)=(x +1)/2.- For x>3, g(x)=(5x -3)/(x +3).Now, let me verify this for x=3:g(3)=(5*3 -3)/(3 +3)=12/6=2, which matches the given definition at x=3.Similarly, for x approaching 3 from above, g(x) approaches (5*3 -3)/(3 +3)=12/6=2, which matches the definition at x=3.For x approaching 2 from below, g(x)=6*2/(2 +6)=12/8=1.5, which matches g(2)=1.5.For x approaching 2 from above, g(x)=(2 +1)/2=1.5, which also matches.So, the function is continuous at x=2 and x=3.Therefore, the complete definition of g(x) is:- g(x)=6x/(x +6) for x in [0,2),- g(x)=(x +1)/2 for x in [2,3],- g(x)=(5x -3)/(x +3) for x>3.Now, let's compute g(2021). Since 2021>3, we use the third case:g(2021)=(5*2021 -3)/(2021 +3)=(10105 -3)/2024=10102/2024.Simplify this fraction:Divide numerator and denominator by 2: 10102 ÷2=5051, 2024 ÷2=1012.So, g(2021)=5051/1012.Wait, can this be simplified further? Let's check GCD(5051,1012).Using Euclidean algorithm:5051 ÷1012=4 with remainder 5051 -4*1012=5051 -4048=1003.Now, GCD(1012,1003).1012 ÷1003=1 with remainder 9.GCD(1003,9).1003 ÷9=111 with remainder 4.GCD(9,4).9 ÷4=2 with remainder 1.GCD(4,1)=1.So, GCD is 1. Therefore, 5051/1012 is in simplest terms.So, g(2021)=5051/1012≈4.9921, which is approximately 5, but exactly 5051/1012.Wait, but earlier when I assumed g(x)=6x/(x +6) for all x, I got g(2021)=12126/2027≈5.98224. But that was incorrect because the function is piecewise defined.So, the correct value is 5051/1012.Similarly, for g(1/2021). Since 1/2021≈0.0004945, which is less than 2, we use the first case:g(1/2021)=6*(1/2021)/(1/2021 +6)=6/(1 +6*2021)=6/(1 +12126)=6/12127.So, g(1/2021)=6/12127.Wait, but earlier when I tried to compute g(1/2021) using the transformation h(x), I got a different result. Let me check that.Wait, in the initial problem, the assistant introduced a function h(x)=1/g(1/x). Then, through some steps, they found h(2021)=2021.5, which led to g(2021)=2021.5.But according to my current approach, g(2021)=5051/1012≈4.9921, which is approximately 5, not 2021.5.This suggests that the assistant's approach might have been incorrect, possibly due to an incorrect assumption about the function h(x).Wait, let me see. The assistant defined h(x)=1/g(1/x). Then, they tried to find h(h(x))= (3x +1)/3.But perhaps this transformation is not valid or leads to incorrect conclusions because the function g is piecewise defined, and h(x) might not follow a simple pattern.Alternatively, maybe the assistant made a mistake in their reasoning.In any case, based on my analysis, the correct expressions for g(x) are piecewise, and thus:g(2021)=5051/1012≈4.9921,g(1/2021)=6/12127≈0.000495.But wait, 5051/1012 is approximately 4.9921, which is very close to 5. Let me check:5051 ÷1012:1012*5=5060, which is 9 more than 5051. So, 5051=1012*5 -9.Therefore, 5051/1012=5 -9/1012≈5 -0.0089≈4.9911.So, approximately 4.9911.Similarly, 6/12127≈0.000495.But let me check if these can be simplified or expressed differently.Wait, 5051/1012 can be written as (5051 ÷ GCD)/ (1012 ÷ GCD). Since GCD is 1, it remains 5051/1012.Similarly, 6/12127 is already in simplest terms.Alternatively, perhaps I can express these as mixed numbers or decimals, but the problem doesn't specify the form, so fractions are fine.Therefore, the answers are:(a) g(2021)=5051/1012,(b) g(1/2021)=6/12127.But wait, let me double-check my earlier steps to ensure I didn't make a mistake.For x>3, g(x)=(5x -3)/(x +3). So, for x=2021:g(2021)=(5*2021 -3)/(2021 +3)= (10105 -3)/2024=10102/2024=5051/1012. Correct.For x=1/2021, which is less than 2, so g(x)=6x/(x +6). Therefore:g(1/2021)=6*(1/2021)/(1/2021 +6)=6/(1 +6*2021)=6/12127. Correct.Yes, these seem correct.But wait, earlier when I assumed g(x)=6x/(x +6) for all x, I got a different result for g(2021). But that was incorrect because the function is piecewise defined, and for x>3, it's (5x -3)/(x +3).Therefore, the correct answers are:(a) g(2021)=5051/1012,(b) g(1/2021)=6/12127.But let me see if these can be simplified further or if there's a pattern.Wait, 5051/1012 is approximately 5, and 6/12127 is approximately 0.000495, which is 1/2021.5.Wait, 6/12127=6/(6*2021 +1)=6/(6*2021 +1). Hmm, not sure if that helps.Alternatively, 6/12127= (6/6)/ (12127/6)=1/2021.1667≈1/2021.1667.But perhaps it's better to leave it as 6/12127.So, final answers:(a) g(2021)=5051/1012,(b) g(1/2021)=6/12127.But let me check if 5051/1012 can be simplified. As we saw earlier, GCD(5051,1012)=1, so it's irreducible.Similarly, 6/12127 is irreducible.Therefore, the answers are:(a) boxed{dfrac{5051}{1012}},(b) boxed{dfrac{6}{12127}}.

💡Okay, so I have this problem where the function ( g(x) ) satisfies the equation ( g(x - y) = g(x)g(y) ) for all real numbers ( x ) and ( y ), and it's also given that ( g(x) ) is never zero for any real ( x ). I need to find ( g(5) ).Hmm, let's start by trying to understand what kind of function ( g(x) ) could be. The equation given is a functional equation, and it reminds me of exponential functions because they have properties where operations in the exponent translate to multiplication or division in the function's output. For example, ( e^{x - y} = e^x e^{-y} ), which is similar but not exactly the same as the given equation.Wait, in the given equation, it's ( g(x - y) = g(x)g(y) ). So if I let ( x = y ), then ( g(0) = g(x)g(x) ), which means ( g(0) = [g(x)]^2 ). That suggests that ( g(0) ) is the square of ( g(x) ) for any ( x ). But ( g(0) ) is a constant, right? So if ( [g(x)]^2 ) is constant, then ( g(x) ) must be either a constant function or a function whose square is constant. But since ( g(x) ) is never zero, it can't be zero anywhere, so it must be a constant function.Wait, is that necessarily true? Let me think again. If ( [g(x)]^2 = g(0) ) for all ( x ), then ( g(x) ) is either ( sqrt{g(0)} ) or ( -sqrt{g(0)} ). But since ( g(x) ) is never zero, it's either always positive or always negative. However, the functional equation ( g(x - y) = g(x)g(y) ) might impose more restrictions.Let me test this idea. Suppose ( g(x) = c ) for some constant ( c ). Then plugging into the equation, we get ( c = c cdot c ), which implies ( c = c^2 ). So ( c^2 - c = 0 ), which gives ( c(c - 1) = 0 ). But since ( g(x) neq 0 ), ( c ) can't be zero, so ( c = 1 ). Therefore, ( g(x) = 1 ) for all ( x ). That would mean ( g(5) = 1 ).But wait, is this the only possibility? I mean, could ( g(x) ) be something else? Let me consider another approach. Maybe I can find ( g(0) ) first. Let me set ( x = y ). Then ( g(x - x) = g(0) = g(x)g(x) ), so ( g(0) = [g(x)]^2 ). As I thought earlier, this suggests that ( g(x) ) is constant because ( [g(x)]^2 ) is the same for all ( x ).But let me try another substitution. Let me set ( y = 0 ). Then the equation becomes ( g(x - 0) = g(x)g(0) ), which simplifies to ( g(x) = g(x)g(0) ). Since ( g(x) ) is never zero, I can divide both sides by ( g(x) ) to get ( 1 = g(0) ). So ( g(0) = 1 ).Okay, so ( g(0) = 1 ). From earlier, ( [g(x)]^2 = g(0) = 1 ), so ( g(x) = pm 1 ). But does ( g(x) ) have to be the same sign for all ( x )? Let's see. Suppose there exists some ( a ) such that ( g(a) = -1 ). Then, using the functional equation, ( g(a - a) = g(0) = g(a)g(a) = (-1)(-1) = 1 ), which is consistent. But what about ( g(a - b) = g(a)g(b) ). If ( g(a) = -1 ) and ( g(b) = -1 ), then ( g(a - b) = 1 ). But if ( g(a - b) = 1 ), does that cause any issues? Let me think.Wait, if ( g(x) ) can be both 1 and -1 depending on ( x ), does that satisfy the functional equation? Let me test it. Suppose ( g(x) = 1 ) for all ( x ). Then ( g(x - y) = 1 = 1 cdot 1 = g(x)g(y) ), which works. Now, suppose ( g(x) = -1 ) for all ( x ). Then ( g(x - y) = -1 = (-1)(-1) = 1 ), which is not equal to -1. So that doesn't work. Therefore, ( g(x) ) cannot be -1 for all ( x ).But what if ( g(x) ) is 1 for some ( x ) and -1 for others? Let's say ( g(a) = -1 ) and ( g(b) = 1 ). Then ( g(a - b) = g(a)g(b) = (-1)(1) = -1 ). But ( g(a - b) ) is either 1 or -1. If it's -1, then that's consistent. But if ( g(a - b) = 1 ), then it's inconsistent. So it's possible only if ( g(a - b) = -1 ), which would require that ( a - b ) is such that ( g(a - b) = -1 ). But this seems too restrictive because ( a ) and ( b ) can be any real numbers, so ( a - b ) can be any real number. Therefore, ( g(x) ) would have to be -1 for all ( x ), which we saw doesn't work because ( g(x - y) = -1 ) but ( g(x)g(y) = 1 ). So that's a contradiction.Therefore, the only possibility is that ( g(x) = 1 ) for all ( x ). Hence, ( g(5) = 1 ).Wait, but let me double-check. Suppose ( g(x) = e^{kx} ) for some constant ( k ). Then ( g(x - y) = e^{k(x - y)} = e^{kx}e^{-ky} = g(x)g(-y) ). But the given equation is ( g(x - y) = g(x)g(y) ). So unless ( g(-y) = g(y) ), which would require ( e^{-ky} = e^{ky} ), implying ( -ky = ky ), so ( k = 0 ). Therefore, ( g(x) = e^{0} = 1 ) for all ( x ). So that also leads to ( g(x) = 1 ).Alternatively, if I consider ( g(x) = a^{x} ), then ( g(x - y) = a^{x - y} = a^{x}a^{-y} = g(x)g(-y) ). Again, unless ( g(-y) = g(y) ), which would require ( a^{-y} = a^{y} ), implying ( a^{2y} = 1 ) for all ( y ), which is only possible if ( a = 1 ). So again, ( g(x) = 1 ) for all ( x ).Therefore, it seems that the only solution is ( g(x) = 1 ), so ( g(5) = 1 ).Another approach: Let me consider the functional equation ( g(x - y) = g(x)g(y) ). Let me set ( x = 0 ). Then ( g(0 - y) = g(0)g(y) ), which simplifies to ( g(-y) = g(0)g(y) ). But earlier, we found that ( g(0) = 1 ), so ( g(-y) = g(y) ). Therefore, ( g ) is an even function.Now, let me consider ( g(x + y) ). Wait, the given equation is for ( x - y ), but maybe I can manipulate it to get an expression for ( g(x + y) ). Let me set ( y = -z ). Then the equation becomes ( g(x - (-z)) = g(x)g(-z) ), which is ( g(x + z) = g(x)g(-z) ). But since ( g(-z) = g(z) ), this becomes ( g(x + z) = g(x)g(z) ). So now, we have ( g(x + z) = g(x)g(z) ) for all ( x, z ).Ah, so ( g ) satisfies the Cauchy functional equation for multiplication, ( g(x + z) = g(x)g(z) ), and it's also given that ( g(x) neq 0 ) for all ( x ). The solutions to this are exponential functions, specifically ( g(x) = e^{kx} ) for some constant ( k ). But earlier, we saw that this leads to ( k = 0 ), so ( g(x) = 1 ).Alternatively, since ( g(x + z) = g(x)g(z) ) and ( g(x) ) is never zero, we can take logarithms. Let ( h(x) = ln(g(x)) ). Then ( h(x + z) = ln(g(x + z)) = ln(g(x)g(z)) = ln(g(x)) + ln(g(z)) = h(x) + h(z) ). So ( h ) satisfies the Cauchy functional equation ( h(x + z) = h(x) + h(z) ). The solutions to this, assuming continuity or some regularity condition, are linear functions ( h(x) = kx ). Therefore, ( g(x) = e^{kx} ). But as before, plugging back into the original equation, we find that ( k = 0 ), so ( g(x) = 1 ).Therefore, regardless of the approach, it seems that ( g(x) = 1 ) is the only solution, so ( g(5) = 1 ).Wait, but in the initial problem, the user provided a solution where they set ( x = 5 ) and ( y = frac{5}{2} ), leading to ( gleft( frac{5}{2} right) = g(5)gleft( frac{5}{2} right) ), and since ( gleft( frac{5}{2} right) neq 0 ), they concluded ( g(5) = 1 ). That seems much simpler. So maybe I can think of it that way too.Let me try that. Let me set ( x = 5 ) and ( y = frac{5}{2} ). Then ( g(5 - frac{5}{2}) = g(5)g(frac{5}{2}) ). Simplifying, ( g(frac{5}{2}) = g(5)g(frac{5}{2}) ). Since ( g(frac{5}{2}) neq 0 ), I can divide both sides by ( g(frac{5}{2}) ) to get ( 1 = g(5) ). So ( g(5) = 1 ).That's a much more straightforward approach. It directly uses the functional equation with specific values to isolate ( g(5) ). So regardless of the general form of ( g(x) ), by choosing ( x = 5 ) and ( y = frac{5}{2} ), we can immediately find ( g(5) ).But wait, does this approach assume that ( g(frac{5}{2}) ) is non-zero? Yes, but the problem states that ( g(x) neq 0 ) for all real ( x ), so that's valid. Therefore, this method is correct and leads directly to ( g(5) = 1 ).So, in summary, whether I approach it by considering the general properties of the function, leading to the conclusion that ( g(x) = 1 ) for all ( x ), or by choosing specific values in the functional equation to isolate ( g(5) ), I arrive at the same answer: ( g(5) = 1 ).

💡First, I need to calculate the time taken to travel from the village to the post-office. The distance is approximately 20 km, and the speed is 25 km/h. By dividing the distance by the speed, I can find the time taken for this part of the journey.Next, I need to determine the time taken to walk back from the post-office to the village. The distance remains the same at 20 km, but the walking speed is 4 km/h. Again, I will divide the distance by the speed to find the return time.Finally, I will add the time taken for both parts of the journey to find the total time spent.

💡First, I need to convert the number -0.000008691 into scientific notation. In scientific notation, a number is expressed as a × 10^b, where 1 ≤ |a| < 10 and b is an integer.To convert -0.000008691, I'll move the decimal point to the right until the number is between 1 and 10. Moving the decimal point six places to the right gives me -8.691.Since I moved the decimal point six places to the right, the exponent for 10 will be -6. Therefore, the scientific notation for -0.000008691 is -8.691 × 10^-6.Comparing this with the given options:A: -8.691×10^-6B: -8.691×10^6C: -8.691×10^-5D: -8.691×10^5The correct answer is option A: -8.691×10^-6.