Problem:
Let f be a linear function for which f(6)−f(2)=12. What is f(12)−f(2) ?
Answer Choices:
A. 12
B. 18
C. 24
D. 30
E. 36
Solution:
Since f is a linear function, its slope is constant. Therefore
6−2f(6)−f(2)=12−2f(12)−f(2), so 412=10f(12)−f(2)
and f(12)−f(2)=30.
\section*{OR}
Since f is a linear function, it has a constant rate of change, given by
6−2f(6)−f(2)=412=3
Therefore f(12)−f(2)=3(12−2)=30.
OR
If f(x)=mx+b, then
12=f(6)−f(2)=6m+b−(2m+b)=4m
so m=3. Hence
f(12)−f(2)=12m+b−(2m+b)=10m=30
The problems on this page are the property of the MAA's American Mathematics Competitions