# 2003 AMC 12B Problems/Problem 9

## Problem

Let $f$ be a linear function for which $f(6) - f(2) = 12.$ What is $f(12) - f(2)?$ $\text {(A) } 12 \qquad \text {(B) } 18 \qquad \text {(C) } 24 \qquad \text {(D) } 30 \qquad \text {(E) } 36$

## Solution 1

Since $f$ is a linear function with slope $m$, $$m = \frac{f(6) - f(2)}{\Delta x} = \frac{12}{6 - 2} = 3$$ $$f(12) - f(2) = m \Delta x = 3(12 - 2) = 30 \Rightarrow \text (D)$$

## Solution 2

Since $f$ is linear, we can easily guess and check to confirm that $f(x)=3x$. Indeed, $f(6)-f(2)=3(6-2)=12$. So, we have $f(12)-f(2)=3(12-2)=30 \Rightarrow \text (D).$

Solution by franzliszt

## See Also

 2003 AMC 12B (Problems • Answer Key • Resources) Preceded byProblem 8 Followed byProblem 10 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 All AMC 12 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. Invalid username
Login to AoPS