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1. (A graphing question)

$f(x) = \begin{cases} 0, & \text{if $x$ is rational} \\ 1, & \text{if $x$ is irrational} \end{cases}$

I'm not exactly sure how to graph this. I'm thinking that it would be two straight lines of points, but I'm not sure if I'm on the right track. Any explanation would be helpful.

2. (A multiple choice)

Let a and b be nonzero real numbers, then the square root of $a^2 - b^2$ is equal to

  • [a] $a-b$
  • [b] $\pm(a-b)$
  • [c] $a \pm b$
  • [d] $\pm(a \pm b)$
  • [e] None of these

I was thinking that the correct answer would be e) None of these, but again, I'm not sure. Explanations would be appreciated.

Thank you!

Community
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Jordan
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  • If you are interested in how one might modify the first function to be somewhat graphable, try this article: http://mathworld.wolfram.com/DirichletFunction.html – fixedp Aug 23 '14 at 15:30
  • For 1., see also http://math.stackexchange.com/questions/146078/graph-of-the-dirichlet-function – Cookie Aug 23 '14 at 15:35
  • Please do not post unrelated question within one. A related issue is that the title is not meaningful. The fact that something was assigned to you in the summer says nothing about the content. And "AP Calculus AB" is not something that most people outside of the U.S. know about. –  Aug 23 '14 at 15:46
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    for #2, why don't you just try (square) each of them. Also, you should be able to recognize $(a + b)(a - b)$. –  Aug 23 '14 at 16:19

3 Answers3

1

  1. It is impossible to graph this function, although it would appear to be two straight lines of points $y=0$ and $y=1$.

  2. Try putting some values in to the various choices and see what you get. For example $\sqrt{9}$ doesn't give you two values $\pm 3$. So, you are correct the answer is "e".

Paul Sundheim
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For the second question, try squaring each of the possible answers and see if any give you $a^2-b^2$.

candido
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The thing to observe for question 2 is that, for any x, the product of any two elements of the set of square roots of x is x only if those elements are the same. More formally, $$\forall x,\,S=\{y:y^2=x\},\,a,b\in S:a\times b=x\iff a=b$$

While this may seem obvious, or even uselessly trivial, expressing such notions as actual mathematical statements can allow us to use them explicitly.

This means that you can consider the cases proposed in each choice independantly. The choices given are:

[a] $a−b$
[b] $\pm(a−b)$
[c] $a\pm b$
[d] $\pm(a\pm b)$

But these are merely shorhand for the unions of one or more of these propositions:

$(a+b)^2 \stackrel{?}{=} a^2−b^2$
$(a-b)^2 \stackrel{?}{=} a^2−b^2$
$(-a+b)^2 \stackrel{?}{=} a^2−b^2$
$(-a-b)^2 \stackrel{?}{=} a^2−b^2$

All of those propositions are false for at least some value of $a,b$. Therefore none of answers a - d can be correct.

AJMansfield
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