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))?
Answers
MathMate
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.
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.
Steve
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
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
maria
f(x)=3/2x-4022