# Composition of Two Functions

• May 14th 2011, 10:44 AM
topsquark
Composition of Two Functions
This is a question by PapaSmurf:
Quote:

2)

Let and f(x) = 9x + k and g(x) = kx + 9.
Find the product of all distinct k such that f(g(x)) = g(f(x)). [Word for word, no typos]
• May 14th 2011, 10:55 AM
TheEmptySet
Given that

$f(x)=9x+k \quad g(x)=kx+9$

then

$f(g(x))=9(kx+9)+k \quad \text{ and } \quad g(f(x))=k(9x+k)+9$

setting them equal and expanding gives

$9kx+81+k=9kx+k^2+9 \iff k^2-k-72=0$

Now just solve for k and multiply the two solutions together.
• May 14th 2011, 11:06 AM
PapaSmurf
Awesome, thank you :)

But, I don't completely understand what you mean at the end.
Quote:

Now just solve for k and multiply the two solutions together.
I get $k = 9, -8$
but don't understand where to go from there... would that be my answer?
• May 14th 2011, 11:26 AM
TheEmptySet
In the original problem statement is says find the product all of distinct k.

Well you have the two distinct values of k so find their product so multiply them together.