answer:1. **Problem Setup:** Given the point set ( left{(x, y) mid y=(-1)^{k} sqrt{1-(x-2k)^{2}}, 2k-1 leq x leq 2k+1, k in mathbf{Z} right} ), we are asked to: 1. Plot the graph of this set. 2. Consider it as the graph of a function ( y = f(x) ). Prove that: [ f(x+4) = f(x), quad text{and} quad f(x+2) + f(x) = 0. ] 2. **Part (1): Graphing the Point Set:** First, note the intervals ( I_k = [2k-1, 2k+1] ) for each ( k in mathbf{Z} ). - When ( k ) is an even number, the point set is composed of a sequence of semicircles above the ( x )-axis. - When ( k ) is an odd number, the point set is composed of a sequence of semicircles below the ( x )-axis. These semicircles' endpoints meet at the ( x )-axis, forming a continuous curve. Here is a rough sketch of these semicircles (for illustration, you should plot this graph). 3. **Part (2): Proving the Functional Equations:** Given ( f(x) = (-1)^{k} sqrt{1 - (x - 2k)^{2}} ) for ( 2k-1 leq x leq 2k+1 ): **Step 1:** Show ( f(x+4) = f(x) ). - Observe that ( x+4 ) lies in the interval ( 2(k+2)-1 leq x+4 leq 2(k+2)+1 ): [ 2(k+2)-1 leq x+4 leq 2(k+2)+1 ] - This can be rearranged as: [ 4k + 7 leq x+4 leq 4k + 9 ] - This interval can be transformed back to the original form for ( x ): [ 2k-1 leq x leq 2k+1 ] - Hence: [ f(x+4) = (-1)^{k+2} sqrt{1 - [(x+4) - 2(k+2)]^{2}} ] - Simplifying the expression inside: [ (x+4) - 2(k+2) = x - 2k ] - Therefore: [ f(x+4) = (-1)^{k+2} sqrt{1 - (x - 2k)^{2}} ] - Since ( (-1)^{k+2} = (-1)^{k} ) (because raising -1 to any even power gives 1): [ f(x+4) = (-1)^{k} sqrt{1 - (x - 2k)^{2}} = f(x) ] **Step 2:** Show ( f(x+2) + f(x) = 0 ). - Observe that ( x+2 ) lies in the interval ( 2(k+1)-1 leq x+2 leq 2(k+1)+1 ): [ 2(k+1)-1 leq x+2 leq 2(k+1)+1 ] - This can be rearranged as: [ 4k + 3 leq x+2 leq 4k + 5 ] - This interval also transforms similar to the previous argument: [ 2k-1 leq x leq 2k+1 ] - Hence: [ f(x+2) = (-1)^{k+1} sqrt{1 - [(x+2) - 2(k+1)]^{2}} ] - Simplifying the expression inside: [ (x+2) - 2(k+1) = x - 2k ] - Therefore: [ f(x+2) = (-1)^{k+1} sqrt{1 - (x - 2k)^{2}} ] - Knowing that ( (-1)^{k+1} = -(-1)^{k} ): [ f(x+2) = -(-1)^{k} sqrt{1 - (x - 2k)^{2}} = -f(x) ] - Hence: [ f(x+2) + f(x) = -f(x) + f(x) = 0 ] 4. **Conclusion:** The function (f(x)) is periodic with period 4 and satisfies ( f(x+4) = f(x) ) (proving periodicity) and ( f(x+2) + f(x) = 0 ) (demonstrating specific periodic behavior). [ boxed{f(x+4) = f(x), quad f(x+2) + f(x) = 0} ]

answer:From the problem, we know that xi can take values of 2, 3, or 4. The probability of xi=2 is P(xi=2)=frac{2}{5} times frac{1}{4} = frac{1}{10}. The probability of xi=3 is P(xi=3)=frac{2}{5} times frac{3}{4} times frac{1}{3} + frac{3}{5} times frac{2}{4} times frac{1}{3} + frac{3}{5} times frac{2}{4} times frac{1}{3} = frac{3}{10}. The probability of xi=4 is P(xi=4)=1-frac{1}{10}-frac{3}{10}=frac{6}{10}. Therefore, the expected value of xi is Exi=2 times frac{1}{10} + 3 times frac{3}{10} + 4 times frac{6}{10} = boxed{frac{7}{2}}. Thus, the answer is B. To solve this problem, we need to understand the possible values of xi and calculate their respective probabilities. Then, we can find the expected value of xi. This is a medium-difficulty problem, and it is important to carefully read the problem and properly apply the multiplication rule for independent events.

answer:To find out how many hours James rode, we can use the formula: Time = Distance / Speed In this case, the distance is 80 miles and the speed is 16 miles per hour. So we can plug these values into the formula: Time = 80 miles / 16 miles per hour Time = 5 hours So, James rode for boxed{5} hours.

answer:Cory bought 1 patio table and 4 chairs. The cost of the patio table is 55. The cost of each chair is 20. The total cost for the chairs is 4 chairs * 20/chair = 80. The total amount Cory spent on the patio table and chairs is the cost of the table plus the cost of the chairs: 55 (table) + 80 (chairs) = 135. Cory spent boxed{135} altogether on the patio table and chairs.