Question

If f and g are linear functions such that g(f(x)) = 2x + 6, and the graph of y = f(g(x)) passes through the origin, what is the value of f(g(2011))?

Answer 1

The question does not say that the coefficients of g(x) and f(x) are integers, so there may be many possible solutions.

One such solution is
f(x)=x+3
and
g(x)=2x
There is no constant term in g(x) because it passes through the origin.

This way g(f(x))=g(x+3)=2(x+3)=2x+6
and g(x) passes through the origin.

For this solution,
g(2011)=4022.

Answer 2

One solution is

f(x) = x+6
g(x) = 2x-6

so,

g(f) = 2f-6 = 2(x+6)-6 = 2x-6

Thus,

f(g) = g+6 = 2x
check:

f(g(0)) = f(-6) = -6+6 = 0
so f(g(x)) passes through (0,0)

f(g(2011)) = 2(2011) = 4022

Answer 3

f(x)=3/2x-4022