<p>Let the function f be defined by f(2k) = 2f(k) and f(5) = 10. Which of the following functions could be the definition of f(x)?</p>
<p>A) f(x) = x+5
B) f(x) = x-5
C) f(x) = 15 - x
D) f(x) = 2x
E) f(x) = x</p>
<p>I don't really understand the relationship given by f(2k) = 2 f(k).</p>
<p>I tried using this:
If f(x)= 2x, 2f(x) is 4x.</p>
<p>Since f(x) = 2f(x)
2x = 4x</p>
<p>Therefore f(2k) = 2(2k) = 4k</p>
<p>However, how do you derive f(x) = 2x from the above working?</p>
<p>Thanks for helping!</p>
<p>Yes! How did you arrive at D? This is supposed to be an “easy” question but I got stuck.</p>
<p>The problem presents two requirements for the function we are looking for. These requirements do not define one and only one function, but they do give you enough information to choose from the answers.</p>
<p>The easier requirement to understand is the second one: f(5) = 10. Just plug in 5 to each of the answers and rule out the ones that are not 10.</p>
<p>The other requirement is harder: f(2x) = 2f(x)</p>
<p>This is a property that will be satisfied by linear functions which pass thru the origin. The only one that still works is f(x) = 2x – here’s why:</p>
<p>If f(x) = 2x, then f(2x) = 4x</p>
<p>but 4x = 2 times 2x = 2f(x).</p>
<p>In fact, any function of the form f(x) = mx would have met the requirement. So choices d and e would both work – but only d also meets the requirement that f(5) = 10.</p>
<p>Also, the question is slightly mis-worded (enough to make me suspect that it is not college board).</p>
<p>This may seem like nitpicking but:</p>
<p>“Let the function f be defined by 2f(x)=f(2x)” – but that statement is NOT the defining statement of the function. It is a PROPERTY that the as-of-yet-undefined function has. It does not uniquely define the function.</p>
<p>Not a big deal, I know, but ETS does a better job writing questions (except occasionally on a weird QOD).</p>
<p>Thanks! So its to plug in instead of working on the function.</p>
<p>Let the function f be defined by f(2k) = 2f(k) and f(5) = 10. Which of the following functions could be the definition of f(x)?</p>
<p>A) f(x) = x+5
B) f(x) = x-5
C) f(x) = 15 - x
D) f(x) = 2x
E) f(x) = x</p>
<p>f(5)=10 means that when x=5, f(x) must equal 10. Plugging a 5 in for x in the answer choices yields</p>
<p>A) 10
B) 5
C) 10
D) 10
E) 5</p>
<p>So we can eliminate choices (B) and (E). If you can’t understand the rest of the problem, take a guess among (A), (C) and (D).</p>
<p>Now we substitute k=5 into the expression f(2k) = 2f(k). </p>
<p>S0 f(10) = 2f(5) = 2*10 = 20. So when x=10, f(x) must equal 20. Plugging a 10 in for x in the answer choices yields</p>
<p>A) 15
C) 5
D) 20</p>
<p>So we can eliminate choices (A) and (C). Thus, the answer is choice (D).</p>