if f(2a - b) = f(a) * f(b) for all a and b and the function is never equal to 0 find the value of f(5)

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TheGreatGuySP

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If f(2a - b) = f(a) * f(b) for all a and b and the function is never equal to 0 find the value of f(5).

I can't seem to find what to substitute in, I've tried 3 and 1 but cannot figure out how to simplify the right side of the equation. Any help is appreciated, thanks.
 
2a-b =5, is that what you can't figure out? That is just a line. You pick a value for a, double it and decide what to subtract from it to get 5. For example if you pick a to be 3, then you get 6 after you double a, so subtract 1 to get 5. That is a=3 and b=1. Is this what you are having trouble with?
 
I looked at this a bit more closely.
What do you get if a=b=5?
 
Last edited:
TheGreatGuySP said:
If f(2a - b) = f(a) * f(b) for all a and b and the function is never equal to 0 find the value of f(5).

I can't seem to find what to substitute in, I've tried 3 and 1 but cannot figure out how to simplify the right side of the equation. Any help is appreciated, thanks.
Click to expand...
What if you just let a=b=5?

In fact, for more general information, what if a=b=x?
 
Has anyone gotten one answer for f(5). I get two values and can't seem to get down to just one value.
 
So there is no solution(?) as f(x) will not be a function. I am assuming that the domain is all real numbers as nothing was mentioned about the domain.
 
Dr.Peterson said:
What if you just let a=b=5?

In fact, for more general information, what if a=b=x?
Click to expand...
You got into my head and figured out to use a=b=5 before I edited my post. How can you see things in my brain before I do? I know that I am slow but how can you use my brain faster than I can. This is not possible! Who exactly are you??!!
 
In fact, the only function satisfying this is the constant function f(x)= 1 for all x!

I would start by looking at special values. f(2a- b)= f(a)f(b) so if a= b= 0, f(0)= f(0)f(0). Either f(0)= 0 or f(0)= 1. Since "the function is never equal to 0", f(0)= 1. If we take b= 0, f(2a)= f(a)f(0)= f(a). Now take a= x, b= 2x. Then f(2a- b)= f(2x- 2x)= f(0)= 1= f(x)f(2x)= (f(x))^2 so f(x) is either 1 or -1 for all x! Since f(0)= 1, f(x)= 1 for all x. In particular, f(5)= 1.
 
Why was I thinking that x=-1 is a solution to x=x^2! In the corner for 1hour.
 
Dr.Peterson said:
I suggest you go there for -1 hour. Just start counting: 1, 2, 3, ... and come out when you get to -1.

On second thought, it's such a serious error, I'm going to add 1 hour to your sentence ...
Click to expand...
I think that your sentence is more than fair for this mistake.
 
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