Question

f is a function defined on the whole real line which has the property that f (1 + x) = f (2-x) for all x. Assume that the equation f(x) = 0 has 8 distinct real roots. Find the sum of these roots.

Answer 1

f(1+x) = f(2-x) for all x ----- (1)
Let 'a' be one of the eight distinct real roots. Then f(a) = 0.
Choose x such that 1+x = a or x = a-1. Plug this x value into (1):
f(1+a-1) = f(2-a+1)
f(a) = f(3-a)
Since f(a) = 0, f(3-a) = 0 which makes (3-a) another root.
The sum of these two roots = a + 3-a = 3.

Answer 2

Similarly, if 'b' is another distinct root, then (3-b) will also be a root.

Therefore, the 8 distinct roots add up to: 4*3 = 12.

Answer 3

How do you explain the fact that the sums of the other 3 pairs of roots is also 3? How do you know that the roots of this function come in pairs?

Answer 4

I hope you didn't pick on Ask dr Math...

Answer 5

Choose another distinct root 'b'. Use the same argument as in my first reply. If 'b' is a root, then (3-b) is also a root and these two roots will also add to 3.
Then pick another distinct root 'c' and later 'd' and use the same argument.

Answer 6

It is given that f(1+x) = f(2-x) for ALL x.
Add the two arguments: (1+x) + (2-x) = 3.
If you give me ANY argument 'arg', I can quickly find its counterpart, which is, 3-arg
making f(arg) = f(3-arg).
For example, I can conclude f(2513) = f(-2510) because 2513 + (-2510) = 3.

Thus, if 'a' is a root, then (3-a) has to be another root because they add up to 3.

Answer 7

I guess it works in this case because they are 8 distinct roots. Will it work for 7?

Answer 8

The very nature of this problem says, for every distinct (1+x) there is a corresponding (2-x) that will make f(1+x) = f(2-x). That makes them occur in distinct PAIRS. So the number of roots a function with the property f(1+x) = f(2-x) will be an even number because they occur in pairs.

Answer 9

It did give me a headache ... Thanks !

Answer 10

One exception to my above statement is when x = 0.5 which gives the same value for (1+x) and (2-x). Thus if 1.5 is one root, then the other root is 3-1.5 = 1.5 and therefore they don't give two distinct roots. For ALL other values of x, (1+x) and (2-x) will always be different forming a distinct pair.

So if this function had 7 distinct roots, then one of the roots has to be 1.5 and the other six roots form 3 distinct pairs that add to 9. This the sum of the 7 roots will be 9+1.5=10.5