I don't understand the college board's explanation of this answer

Test Preparation SAT Preparation
1

<p>Ok so the question is </p>

<p>A certain function f has the property that f(x+y) = f(x) + f(y) for all values of x and y. Which of the following statements must be true when a = b?</p>

<p>I. f(a+b) = 2f(a)
II. f(a+b) = [f(a)]^2
III. f(b) + f(b) = f(2a)</p>

<p>I know the answer, I understand it's I and III. I know that II cannot be true because if a were to equal 3, then f(3+b)= 6 since a=b. However [f(a)]^2 would be f(3+0)^2=9. </p>

<p>But, this is the extra long, supposedly "simpler" way the college board justifies why II cannot be always true.</p>

<p>When a=b, statement II, f(a+b)=f[(a)]^2, becomes f(a+a)=f(2a)=[f(a)]^2. (I understand all of this so far) This statement is not necessarily true. For example, if f(x)=3x for all values of x </p>

<p>(here is where I get lost)</p>

<p>then f(x+y)=3(x+y)=3x+3y=f(x)+f(y) for all values of x and y; </p>

<p>(First of all, if f(x)=3x, then wouldn't the equation f(x+y) become f(3x+y)?? Why are they replacing the F with THREE and not the X inside of the function f(x+y) with 3X?!?!)</p>

<p>however, if a=4 then f[(2)(4)]=(3)(2)(4)=24. </p>

<p>(I understand the f[(2)(4)] because as we stated earlier, f(a+b)=f(a+a)=f(2a) ) </p>

<p>but, [f(4)]^2 = [(3)(4)]^2 = 12^2 = 144. Therefore, f(a+a)=[f(a)]^2 does not hold for all values of a.</p>

<p>I'm sorry if this is confusing, but if anyone understands can they please clarify?! I mean *** is wrong with the college board, or am I just stupid?!</p>

2

<p>My explanation is that , if f(x+y) = f(x) + f(y), then f(a+b) = f(a) + f(b), which would equal either f(2a), 2f(a), f(2b), 2f(b), NOT [f(a)]^2, which would only be true if a equaled 2. I dont understand *** the college board is trying to tell me.</p>

3

<p>What…? I don’t even…
You’re over thinking it. I wouldn’t even plug in numbers. First of all you don’t know what f(x) actually is. all you know is that f(a+b) = f(a) + f(b) and that a = b.</p>

<p>so in the function f(a+b) = f(a) + f(b), you can pretty much use a and b interchangeably… </p>

<p>I and III are obviously true then. II is not true because then f(a+b) would have to equal f(a) * f(a) which it doesn’t.</p>

<p>Don’t over complicate things</p>

4

<p>They are introducing the function f(x)=3x as an example of a function that satisfies the original requirement: f(x+y)=f(x) + f(y)</p>

<p>It confused you becasue they used the letter ‘x’ while defining the function and also while describing its special property. Instead of saying f(x+y) = f(x) + f(y), they could have said it all in words, such as:</p>

<p>We are looking for a function that has a special property: if you inut a sum into the function, the output is equal to the sum of what you get when you input the two values separately and then add the ouputs. [It is more concise when you say it algebraically, as they did!]</p>

<p>Now we are going to give you an example of a function that has that property: the tripling function. Try finding a sum and then tripling it. Then try first tripling the two numbers and THEN finding the sum. You get the same answer. Thus, the tripling function, f(x) = 3x, meets the original stated requirements.</p>

<p>OK, now that we have you convinced that this function meets the special requirement, we’ll show you that it does NOT satisfy rule II. Let a = 4. f(2a) = f(8) = 24 but f(4)^2 = 12^2 = 144.</p>

<p>So rule II does NOT have to be obeyed by EVERY function that meets the original requirement!</p>

<hr>

<p>But it’s still pretty nasty…</p>

5

<p>The requirement f(x+y)=f(x) + f(y) can also be stated more succinctly:</p>

<p>“The output of the sum is equal to the sum of the outputs.”</p>

<p>This is true for all linear functions of the form y=kx.</p>

<p>But it’s also NEARLY true for some non-linear functions over portions of their domain. The one that comes to mind is the sine function. In general, as a non-linear function, the sine function does not satisfy this property. For example, it’s easy to see that</p>

<p>sin(30) + sin(30) =/= sin (30+30)</p>

<p>But for small angles, the sine graph is linear-ish! So if you try it on your calculator, you will see that for example, sin(2) + sin(3) is not far from sin(5). [In degrees…]</p>

<p>So I’ll leave this with two questions to play with:</p>

<ol>
<li><p>What is the largest value of x – measured in degrees – such that sin(x) + sin(x) is within 5% of sin(2x). [This can be done by trial and error, but there is a better way.] (</p></li>
<li><p>What other functions can you think of that are linear-ish in this way?</p></li>
</ol>

<p>And again, this is for nerd-points only and clearly not on the SAT :)</p>

6

<p>Ok I get what they did pckeller, but why would they go way out of the way and show you the tripling function to show that rule II is not always true?!?! The collegeboard seems idiotic when it comes to teaching at times. Also, I still don’t understand this part</p>

<p>f(x+y)=3(x+y)=3x+3y=f(x)+f(y)</p>

<p>are they trying to say that 3x+3y would be the same as f(x)+f(y)?</p>

<p>I understand that 3(x+y) will also equal 3x+3y, but then why did they throw that f(x)+f(y) in there, just to show that the tripling function applies to f(x)+f(y)??</p>

7

<p>and for your question, I don’t want to do it by trial and error, but i’m really trying to figure this out!! You said “small angles” well, anything below 90 is considered a small angle, but numbers that large wouldn’t work, hmmmm</p>

8

<p>They used the tripling function because it’s linear. So the sum of the outputs will be the same as the output of the sum.</p>

<p>Call the inputs x and y. The sum is x+y. And f(x+y) = 3(x+y) – that’s the ouput of the sum.</p>

<p>Now instead, take x and y and put them into the function:
f(x) = 3x
f(y) = 3y.
Now add those ouputs. You get 3x + 3y. </p>

<p>But 3x + 3y is equal to 3(x+y). So the sum of the ouputs was in fact equal to the output of the sum.</p>

<p>To be mathematically rigorous, they had to go through that whole argument just to prove that the tripling function “fit”. Then they show that it does NOT meet requirement #2…</p>

<p>I have to say, this is a particularly challenging function problem. </p>

<p>As for the puzzle I gave…well, let’s just say that there is a trig identity that would help solve it…and let me say again, it’s just for fun – no SAT relevance at all.</p>

9

<p>Oh well I haven’t taken trig yet so I guess i’ll have to do trial and error, is it 23??!</p>

10

<p>To the nearest whole #, it’s 18 degrees…</p>

<p>When you get to trig, you will lean an identity called the double angle identity for sines:</p>

<p>sin(2x)= 2sin(x)cos(x)</p>

<p>So, strictly speaking, the sine function never satisfies the f(x+y) = f(x) + f(y) requirement. But for small angles, cos(x) is close to 1…in fact cos(18)=.95 or so… so at an angle of 18 degrees, you get approximately a 5% difference when you ignore the cosine term. One way to think about this identity is to think of the cosine term as a correction factor, correcting for the fact that sine function is not linear.</p>