Suppose the graph of y=f(x) includes the points (1,5), (2,3), and (3,1).

based only on this information, there are two points that must be on the graph of y=f(f( if we call those points (a, b) and (c, d), what is ab+cd?

17

How I got that

(2,3) and (3,1) have '3' in common in that the y value of the first pairs with the x value of the second.

If you picture a chain, then you start with x = 2, move to y = 3, then move to x = 3 and then y = 1

2 ---> 3 ---> 3 ---> 1

So f(f(2)) = f(3) = 1

If g(x) = f(f(x)), then we know (2,1) is on the graph of g(x)

Repeat for (3,1) and (1,5)

3 ---> 1 ---> 1 ---> 5

f(3) = 1

g(x) = f(f(x)) = f(f(3)) = f(1) = 5

We know that (3,5) is on the graph of g(x)

The two points on g(x) are: (2,1) and (3,5)

Comparing that to (a,b) and (c,d) we can see

a = 2, b = 1, c = 3, d = 5

a*b + c*d = 2*1 + 3*5 = 2 + 15 = 17