#1**0 **

Sorry, but without a single solution such as for example f(0) = 1 this cannot be written in another way.

Perhaps I like Serena or Melody know something different then me about this, but it might take a while before they look at this.

Werter_380:f(f(x)) = 13 - 2(f(x))

Decide f(f(x))

Sorry, but without a single solution such as for example f(0) = 1 this cannot be written in another way.

Perhaps I like Serena or Melody know something different then me about this, but it might take a while before they look at this.

reinout-g Jan 29, 2014

#2**0 **

Thanks reinout-g

I actually looked probably when you did.

I've got no good ideas.

I don't think it is solvable but i could be wrong.

Sorry.

I actually looked probably when you did.

I've got no good ideas.

I don't think it is solvable but i could be wrong.

Sorry.

Melody Jan 30, 2014

#3**0 **

Hey Werter!

I see someone referred to me. Suggesting that I may have a solution.

Well... I do not have a solution either.

Buuuuut... let's see what we can say...

There is more than 1 solution, so we need indeed some extra condition.

Suppose for instance that f(x)=c for some constant c.

Then the equation becomes:

f(f(x)) = 13 - 2 f(x)

f(c) = 13 - 2 c

c = 13 - 2c

3c = 13

c = 13 / 3

So the corresponding solution is f(x) = 13 / 3.

Or suppose f(x) = ax + b with non-zero a.

Then:

f(f(x)) = 13 - 2 f(x)

f(ax + b) = 13 - 2 (ax + b)

a (ax + b) + b = 13 - 2 (ax + b)

(a^{2} + 2a) x = 13 - ab - b - 2b ................. using Melody's hint to use [sup]

a (a + 2) x = 13 - (a + 3) b

Since this equation should hold for any x, both the left side and the right side will have to be zero.

Since we started with a non-zero "a", we need a = -2 and b = 13.

So another solution is f(x) = -2x + 13.

I see someone referred to me. Suggesting that I may have a solution.

Well... I do not have a solution either.

Buuuuut... let's see what we can say...

There is more than 1 solution, so we need indeed some extra condition.

Suppose for instance that f(x)=c for some constant c.

Then the equation becomes:

f(f(x)) = 13 - 2 f(x)

f(c) = 13 - 2 c

c = 13 - 2c

3c = 13

c = 13 / 3

So the corresponding solution is f(x) = 13 / 3.

Or suppose f(x) = ax + b with non-zero a.

Then:

f(f(x)) = 13 - 2 f(x)

f(ax + b) = 13 - 2 (ax + b)

a (ax + b) + b = 13 - 2 (ax + b)

(a

a (a + 2) x = 13 - (a + 3) b

Since this equation should hold for any x, both the left side and the right side will have to be zero.

Since we started with a non-zero "a", we need a = -2 and b = 13.

So another solution is f(x) = -2x + 13.

I like Serena Jan 30, 2014