Question

A question from a contest.

Answer 1

I know that f(b) = 0 then.. :(

Answer 2

The first thing you need to look at, is the term \(f(f(0))=0\). So \(f(b)=0\). This means that \[ab+b=0 \Longrightarrow ab=-b\]Hence, \(a=-1\).

Answer 3

Yeah, my bad. a = -1. As pointed out above.

Answer 4

How do you conclude that if f(b) = 0, ab + b = 0 D:

Answer 5

We conclude that from the definition of the function. \[f(x)=ax+b\]

Answer 6

Now focus on \(f(f(f(4)))=9\) If you expand this out, you get that \[f(f(-4+b))=9\]\[f(-(-4+b)+b)=f(4-b+b)=f(4)=9\]\[-4+b=9\]Hence, you have that \(b=13\).

Answer 7

Now that we have the function, we just have to find that last part.\[f(f(f(f(10))))=f(f(f(-10+13)))=f(f(f(3)))\]\[f(f(f(3)))=f(f(-3+13))=f(f(10))\]\[f(f(10))=f(-10+13)=f(3)\]\[f(3)=-3+13=10\]Thus, we conclude that \(f(f(f(f(10))))=10\)

Answer 8

Ahh, thanks! :DD

Answer 9

You're welcome.