# Functions

How do medical breakthroughs happen?  How do you feel about animal testing?  A few dogs that were once paralyzed are now walking again with the aid of some cells from healthy dogs.  Scientists recognize that they are not ready to apply their findings to humans with spinal cord injuries, but they are hopeful about the future.  This development would make a great current event example for your SAT essay.  If you choose to use it as a current event, take notes detailing the facts in this article and how you could apply them to a wide variety of topics.

## Algebra: Functions

Read the following SAT test question and then select the correct answer.

Take the time to read math questions carefully because this will save you from wasting time as you solve the problem.  Start by identifying the bottom line and assessing your options for reaching it.  Choose the most efficient method to attack the problem.  When you think you have an answer, loop back to make sure that it matches the bottom line.
• f(2n) = 2f(n) for all integers n
• f(4) = 4
If f is a function defined for all positive integers n, and f satisfies the two conditions above, which of the following could be the definition of f?

Bottom line: f(n) = ?

Assess your options:  You could start by plugging in f(4) to see which of your answer choices results in the number 4, then check any that do against the first condition.  This is the method recommended by collegeboard.com.  However, this requires multiple steps as you compare each answer choice to both conditions.  Instead, take a moment to think logically about the two conditions that you are given.

Attack the problem:  Start with the second condition because it is already in a format that is easy to use.  If f(4) = 4 and you can use the same variable to represent numbers that are the same,  that is the same as saying that f(n) = n.  Now look at the first condition and think about it logically.  If you multiply the variable within the function by 2, that gives you the same number as multiplying the result of the function by 2.  In order for those two numbers to be the same, the final result of the function has to match the number that is plugged into the function.  In other words, f(n) = n.  Both conditions give you the same definition of the function.

Loop Back:  You found a simple way to define both of the conditions for f(n), so look down at your answer choices.

(A)  f(n) = n - 2
(B)  f(n) = n
(C)  f(n) = 2n
(D)  f(n) = 4
(E)  f(n) = 2n – 4

On sat.collegeboard.org, 46% of the responses were correct.

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