answer：First, I need to find the value of ( g(3) ) using the function ( g(x) = x^3 ).Calculating ( g(3) ):[g(3) = 3^3 = 27]Next, I substitute ( g(3) = 27 ) into the function ( f(x) = 3x - 2 ) to find ( f(g(3)) ).Calculating ( f(27) ):[f(27) = 3 times 27 - 2 = 81 - 2 = 79]Therefore, the value of ( f(g(3)) ) is 79.

answer：First, I need to determine the number of teaching and educational support staff. Given the total number of staff is 200 and there are 24 administrative staff, the combined number of teaching and support staff is 200 - 24 = 176.The ratio of teaching to educational support staff is 10:1. Let’s denote the number of educational support staff as ( x ). Therefore, the number of teaching staff is ( 10x ).Setting up the equation:[x + 10x = 176][11x = 176][x = frac{176}{11} = 16]So, there are 16 educational support staff and ( 10 times 16 = 160 ) teaching staff.Next, to determine how many teaching staff should be included in a sample of 50 using stratified sampling, I calculate the proportion of teaching staff in the total population:[frac{160}{200} times 50 = 40]Therefore, 40 teaching staff members should be included in the sample.

answer：Okay, so I have this problem here where I need to figure out which of the given functions is both an even function and monotonically increasing on the interval (0, +∞). The options are A, B, C, and D. Let me break this down step by step because I want to make sure I understand each part correctly.First, let's recall what an even function is. An even function satisfies the condition f(-x) = f(x) for all x in its domain. This means that if I plug in -x into the function, it should give me the same result as when I plug in x. Graphically, even functions are symmetric about the y-axis. So, I'll need to check each function to see if it meets this criterion.Next, the function needs to be monotonically increasing on the interval (0, +∞). Monotonically increasing means that as x increases, the value of the function also increases. In other words, for any two points x1 and x2 where x1 < x2, f(x1) ≤ f(x2). To determine if a function is increasing, I can look at its derivative. If the derivative is positive on the interval, the function is increasing there.Alright, let's go through each option one by one.**Option A: f(x) = |x| + 2**First, check if it's even. Let's compute f(-x):f(-x) = |-x| + 2 = |x| + 2 = f(x)So, yes, it is an even function because f(-x) equals f(x).Now, check if it's monotonically increasing on (0, +∞). For x > 0, |x| is just x, so f(x) = x + 2. The derivative of this with respect to x is 1, which is positive. Therefore, the function is increasing on (0, +∞).So, option A satisfies both conditions.**Option B: f(x) = 3^{-x}**First, check if it's even. Compute f(-x):f(-x) = 3^{-(-x)} = 3^{x}Compare this to f(x) = 3^{-x}. Clearly, 3^{x} is not equal to 3^{-x} unless x = 0. So, f(-x) ≠ f(x), meaning this is not an even function.Since it doesn't satisfy the even function condition, we can eliminate option B.**Option C: f(x) = √x**First, check if it's even. The function √x is only defined for x ≥ 0, so it doesn't have a domain that includes negative x-values. Therefore, it cannot be an even function because even functions require symmetry about the y-axis, which isn't possible here.So, option C is not an even function and can be eliminated.**Option D: f(x) = -x² + 1**First, check if it's even. Compute f(-x):f(-x) = -(-x)² + 1 = -x² + 1 = f(x)So, yes, it is an even function.Now, check if it's monotonically increasing on (0, +∞). Let's find the derivative:f'(x) = d/dx (-x² + 1) = -2xFor x > 0, -2x is negative. This means the function is decreasing on (0, +∞), not increasing.Therefore, option D is even but not monotonically increasing on the given interval.**Summary:**- Option A: Even and increasing on (0, +∞) – satisfies both conditions.- Option B: Not even – doesn't satisfy.- Option C: Not even – doesn't satisfy.- Option D: Even but decreasing on (0, +∞) – doesn't satisfy.So, the only function that meets both criteria is option A.

answer：First, I need to determine the speed of the train using the formula:[ text{Speed} = frac{text{Distance}}{text{Time}} ]The distance the train travels is equal to its length, which is 700 meters. The time taken to cross the pole is 42 seconds.To find the speed in kilometers per hour (km/hr), I need to convert the distance from meters to kilometers and the time from seconds to hours.Converting 700 meters to kilometers:[ 700 text{ meters} = 0.7 text{ kilometers} ]Converting 42 seconds to hours:[ 42 text{ seconds} = frac{42}{3600} text{ hours} ]Now, I can calculate the speed:[ text{Speed} = frac{0.7 text{ km}}{frac{42}{3600} text{ hours}} ]Simplifying the calculation:[ text{Speed} = frac{0.7 times 3600}{42} text{ km/hr} ][ text{Speed} = frac{2520}{42} text{ km/hr} ][ text{Speed} = 60 text{ km/hr} ]Therefore, the speed of the train is 60 km/hr.

answer：First, I need to understand the relationship between speed and time when the distance remains constant. If the worker walks slower, it will take her longer to reach her office.Given that the worker walks at 4/5 of her normal speed, her new speed is reduced. Since speed and time are inversely proportional, I can use this relationship to find out how much longer it takes her to cover the same distance.I'll start by setting up the proportion between the original speed and the new speed, which will give me the ratio of the new time to the original time.Next, I'll solve for the new time by multiplying the original time by the ratio obtained from the proportion.Finally, by subtracting the original time from the new time, I'll determine how many minutes later than usual the worker reaches her office.

answer：Okay, so I have this matrix N, which is a 2x2 matrix:N = [ [-1, 4], [-6, 3] ]And I need to find scalars r and s such that when I square matrix N, it equals r times N plus s times the identity matrix I. So, the equation is:N² = rN + sIAlright, first things first, I need to compute N squared. To do that, I have to multiply matrix N by itself. Let me recall how matrix multiplication works. Each element of the resulting matrix is the dot product of the corresponding row of the first matrix and column of the second matrix.So, let's compute N² step by step.First, the element in the first row and first column of N² is:(-1)*(-1) + 4*(-6) = 1 - 24 = -23Next, the element in the first row and second column is:(-1)*4 + 4*3 = -4 + 12 = 8Moving to the second row, first column:(-6)*(-1) + 3*(-6) = 6 - 18 = -12And finally, the element in the second row and second column:(-6)*4 + 3*3 = -24 + 9 = -15So, putting it all together, N squared is:N² = [ [-23, 8], [-12, -15] ]Alright, now I need to express N² as rN + sI. Let's write out what rN + sI looks like.First, rN is just r multiplied by each element of N:rN = [ [-r, 4r], [-6r, 3r] ]Then, sI is s multiplied by the identity matrix:sI = [ [s, 0], [0, s] ]Adding these two matrices together, we get:rN + sI = [ [-r + s, 4r + 0], [-6r + 0, 3r + s] ]Simplifying, that's:rN + sI = [ [-r + s, 4r], [-6r, 3r + s] ]Now, we can set this equal to N²:[ [-23, 8], [-12, -15] ] = [ [-r + s, 4r], [-6r, 3r + s] ]This gives us a system of equations by equating corresponding elements:1. From the (1,1) position: -r + s = -232. From the (1,2) position: 4r = 83. From the (2,1) position: -6r = -124. From the (2,2) position: 3r + s = -15Let me write these equations down clearly:1. -r + s = -232. 4r = 83. -6r = -124. 3r + s = -15Looking at equations 2 and 3, they both involve only r. Let me solve equation 2 first:4r = 8Dividing both sides by 4:r = 8 / 4r = 2Alright, so r is 2. Let me check equation 3 to make sure it's consistent:-6r = -12Substituting r = 2:-6*(2) = -12-12 = -12Yes, that's correct. So, r = 2 is consistent with both equations 2 and 3.Now, let's use r = 2 to find s. Let's use equation 1:-r + s = -23Substituting r = 2:-2 + s = -23Adding 2 to both sides:s = -23 + 2s = -21Let me verify this with equation 4 to make sure:3r + s = -15Substituting r = 2 and s = -21:3*(2) + (-21) = 6 - 21 = -15Yes, that works out. So, both equations 1 and 4 are satisfied with r = 2 and s = -21.Therefore, the scalars r and s that satisfy the equation N² = rN + sI are r = 2 and s = -21.**Final Answer**The scalars are boxed{2} and boxed{-21}.**Final Answer**boxed{(2, -21)}