# Spivak chapter 3 question 6

Hello, I have been backtracking recently on Spivak problems to ensure I am fluent and I'm doing this one and I just cannot solve it:

For which numbers C is there a number X such that f(cx) = f(x). Hint: there are a lot more than you might think at first glance.

I get this isn't really calculus but its in the book so I thought the responders here would be most useful.

I appreciate all responses.

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I have moved forward while I wait for a response and have come to problem 9 with characteristic functions and it asks me to express A intersects B, X is either in A or B, and R - A ( all as subsets of C) in terms of $C_A$ and $C_B$.

I don't know what it is asking me to do.

What is $f$? What's its domain? Is it continuous?

If it's continuous and its domain includes zero, those might be useful properties. If there's also a point $x>0$ such that $f(x)=f(0)$, that might also be a useful property.

For the second question, it's asking for algebraic ways of writing the statements given in terms of $C_A,C_B$. For example, the statement $$A\neq\emptyset$$ is equivalent to the expression $C_A\neq 0$.

Thanks. I didn't think of it that way.

pasmith
Homework Helper
Hello, I have been backtracking recently on Spivak problems to ensure I am fluent and I'm doing this one and I just cannot solve it:

For which numbers C is there a number X such that f(cx) = f(x). Hint: there are a lot more than you might think at first glance.
Given c, all you need to do is choose x such that cx = x.

The domain is x cannot equal -1. How does that help me? You could say that it is any number if x=0. Is that acceptable?

For the second problem, I still don't see what you mean. How do I algebraically manipulate them?

I have just realised this is in the wrong space but i don't know how to move it. Im sorry and feel free to move it.

It turned out that C can be anything so i was right there. The second one is a bit tricky though. I will reread the text first.

I see that $C_A$$\bullet$$C_B$ or $C_A$$+$$C_B$ could both be the answer for $C_A\cap C_B$ as both intersect with x. Is this right?How do i tell which it is?

Economics nerd, you're saying that you can say A is not empty by writing $C_A$ is not 0 as, if its zero, x is not in A so A is empty.

I see what you mean but could you please explain to me how these characteristic functions work. I get that C_A(x) means that there is an x for every A. However, I am having relating that to $C_A \cup C_B$?

Any response to my two questions?

Also, in this context, what is the difference between adding, multiplying and subtracting in terms of the domain.

Any thoughts?

Is it normal to have issues with such problems early on, even if it is spivak?

Anybody got any hints?