What Are the Possible Values of C for f(cx) = f(x) to Hold True?

[tridianprime]

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.

 

[tridianprime]

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.

 

[economicsnerd]

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.

 

[economicsnerd]

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 $C_A\neq 0$.

 

[tridianprime]

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

 

[pasmith]

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.

 

[tridianprime]

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?

 

[tridianprime]

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

 

[tridianprime]

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.

 

[tridianprime]

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?

 

[tridianprime]

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 $C_A \cup C_B$?

 

[tridianprime]

Any response to my two questions?

 

[tridianprime]

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

 

[tridianprime]

Any thoughts?

 

[tridianprime]

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

 

[tridianprime]

Anybody got any hints?