With New Problem due Wednsday

[STAR3URY]

Please Help With New Problem...due Wednsday!

The question is the following..

For all x f(x+1) = f(x)+1
a. f(x) has to be unbroken
b. f(x) is non-linear

I just have to come up with ANY example of f(x) for that to be true. So basically come up with any function which is unbroken and is not linear so that f(x+1) = f(x)+1 is true.

One example that he showed us in class was the step function, but he said we can't use that..so we have to make 1 up...can someone please HELP ME?
I though i had it with f(x) = |x^2| , but i am not sure..

f(x^2 + 1) = f(x^2) + 1??

 

[varygoode]

Do you mean f(x)=x^2? Because f(x^2) is not a "function", it's just what value to plug into some function 'f'. You need to find some f(x)="expression" such that f(x+1) = f(x)+1. So your idea, f(x^2), does not answer the question at hand. (And neither does f(x)=x^2.)

Now, I'm unclear what is meant by "unbroken". Do you mean continuous?

 

[STAR3URY]

yes continuous and it has to be a NON-LINEAR function..and it has to follow the rule:

f(x+1) = f(x) + 1

 

[STAR3URY]

anyone ...

 

[HallsofIvy]

Have you given any thought to what your f must be for x an integer?
f(1) can be any number, of course, but then f(2)= f(1+1)= f(1)+ 1. f(3)= f(2+1)= f(1)+ 1+ 1= f(1)+ 2, f(4)= f(3+1)= f(3)+ 1= f(1)+2+1= f(1)+ 3. It's easy to see (or prove by induction) that as long as n is an integer, f(n)= f(1)+ n-1. The LINEAR function f(x)= C+ x-1 for C any constant would work fine but your example must not be a linear. I thought about using the "floor" function but that would not be continuous.

I doubt that there is a continuous, non-linear, function satisfying that!

 

[STAR3URY]

Have you given any thought to what your f must be for x an integer?
f(1) can be any number, of course, but then f(2)= f(1+1)= f(1)+ 1. f(3)= f(2+1)= f(1)+ 1+ 1= f(1)+ 2, f(4)= f(3+1)= f(3)+ 1= f(1)+2+1= f(1)+ 3. It's easy to see (or prove by induction) that as long as n is an integer, f(n)= f(1)+ n-1. The LINEAR function f(x)= C+ x-1 for C any constant would work fine but your example must not be a linear. I thought about using the "floor" function but that would not be continuous.

I doubt that there is a continuous, non-linear, function satisfying that!

My professor said that there are a lot of such functions...

 

[BlackWyvern]

$$n*cos 2 \pi n$$

Would work for integer values. Maybe you heard him wrong and he wants something simple like this.

 

[STAR3URY]

$$n*cos 2 \pi n$$

Would work for integer values. Maybe you heard him wrong and he wants something simple like this.

I can't use any TRIGONOMETRY since we still didnt go over it in class, he wants a regular function.

 

[varygoode]

It's going to be quite the task to find a non-linear, non-periodic, continuous function satisfying $$f(x+1) = f(x)+1$$. You need to find more details about what this "professor" of yours is looking for, or at least make sure what you're telling us is accurate.

Other than that tidbit, good luck on finding this needle in a haystack.