Problem:
A function f has the property that f(3x−1)=x2+x+1 for all real numbers x. What is f(5) ?
Answer Choices:
A. 7
B. 13
C. 31
D. 111
E. 211
Solution:
Let u=3x−1. Then x=(u+1)/3, and
f(u)=(3u+1)2+3u+1+1=9u2+2u+1+3u+1+1=9u2+5u+13
In particular,
f(5)=952+5⋅5+13=963=7
OR
The value of 3x−1 is 5 when x=2. Thus
f(5)=f(3⋅2−1)=22+2+1=7
The problems on this page are the property of the MAA's American Mathematics Competitions