Appearance
question:Below is a portion of the graph of a function, y=u(x): [asy] import graph; size(5.5cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-3.25,xmax=3.25,ymin=-3.25,ymax=3.25; pen cqcqcq=rgb(0.75,0.75,0.75); /*grid*/ pen gs=linewidth(0.7)+cqcqcq+linetype("2 2"); real gx=1,gy=1; for(real i=ceil(xmin/gx)*gx;i<=floor(xmax/gx)*gx;i+=gx) draw((i,ymin)--(i,ymax),gs); for(real i=ceil(ymin/gy)*gy;i<=floor(ymax/gy)*gy;i+=gy) draw((xmin,i)--(xmax,i),gs); Label laxis; laxis.p=fontsize(10); xaxis("",xmin,xmax,Ticks(laxis,Step=1.0,Size=2,NoZero),Arrows(6),above=true); yaxis("",ymin,ymax,Ticks(laxis,Step=1.0,Size=2,NoZero),Arrows(6),above=true); real f1(real x){return -x+3*sin(x*pi/3);} draw(graph(f1,-3.25,3.25),linewidth(1)); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy] What is the exact value of u(-2.33)+u(-0.81)+u(0.81)+u(2.33) ?
answer:Looking at the graph, we can see that u(-2.33) is approximately 2.33, u(-0.81) is approximately 1.19, u(0.81) is approximately -1.19, and u(2.33) is approximately -2.33. Therefore, the exact value of u(-2.33)+u(-0.81)+u(0.81)+u(2.33) is 2.33+1.19+(-1.19)+(-2.33) = boxed{0}. The answer is: 0
question:Let f(x) = 2x - 3 and g(x) = x + 1. What is the value of f(1 + g(2))?
answer:First, we evaluate g(2) by substituting 2 into the function g(x), so g(2) = 2 + 1 = 3. Next, we substitute 3 into the function f(x), so f(1 + g(2)) = f(1 + 3) = f(4). Finally, we substitute 4 into the function f(x), so f(4) = 2(4) - 3 = 8 - 3 = boxed{5}. The answer is: 5
question:The endpoints of a diameter of circle M are (-1,-4) and (-7,6). What are the coordinates of the center of circle M? Express your answer as an ordered pair.
answer:The midpoint of a diameter of a circle is the center of the circle. The midpoint of the diameter with endpoints (-1,-4) and (-7,6) is left(frac{-1+(-7)}{2}, frac{-4+6}{2}right) = left(-frac{8}{2}, frac{2}{2}right) = boxed{(-4,1)}. The answer is: (-4,1)
question:What is the value of binom{8}{4}?
answer:The binomial coefficient binom{n}{k} represents the number of ways to choose k items from a set of n items, where order does not matter. In this case, we want to choose 4 items from a set of 8 items. Using the formula binom{n}{k} = frac{n!}{k!(n-k)!}, we have binom{8}{4} = frac{8!}{4!4!}. Simplifying, we get binom{8}{4} = frac{8times7times6times5}{4times3times2times1} = boxed{70}. The answer is: 70