Functions problem! "If f(k/2)=1 and f(p+1)=9. Show that k^2+p^2 is divisible by 5."

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Helenam

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Not sure how to start this question.
If f(k/2)=1 and f(p+1)=9. Show that k^2+p^2 is divisible by 5.
Any help would be appreciated. Thanks.
 
Helenam said:
Not sure how to start this question.
If f(k/2)=1 and f(p+1)=9. Show that k^2+p^2 is divisible by 5.
Any help would be appreciated. Thanks.
Click to expand...
The statement is not true for an arbitrary function [imath]f[/imath]. For example if [imath]f(x) = x[/imath] then k =2, p = 8 and [imath]k^2+p^2 = 68[/imath], which is not divisible by 5.
What is [imath]f[/imath] in your case?
 
Ah. Now I see. I have been given a function and a point on the graph so I can work it out now. Sometimes I just need a nudge.
 
Helenam said:
Ah. Now I see. I have been given a function and a point on the graph so I can work it out now. Sometimes I just need a nudge.
Click to expand...
I'm curious to see the actual problem, and it might help other readers.

But now you see why we ask you to show the entire problem, not just part of it:

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Sometimes in doing this, you discover what you were missing in your reading of the problem! Nudging yourself isn't a bad idea.
 
Sorry I didn’t think to include it all. The info at the start of the question is as follows.
The graph of the function f(x) =a^x+b, where x is a member of the real numbers and includes the points A(1, -5) and B(2, -3)

Then it asks you to prove the statement as in my initial question.

You’re probably right about the nudging!
Thanks to everyone again.
 
Helenam said:
Sorry I didn’t think to include it all. The info at the start of the question is as follows.
The graph of the function f(x) =a^x+b, where x is a member of the real numbers and includes the points A(1, -5) and B(2, -3)

Then it asks you to prove the statement as in my initial question.
Click to expand...
Thanks. So the actual question can be phrased like this:

Given that f(x) =a^x+b, with f(1) = -5 and f(2) = -3, show that if f(k/2)=1 and f(p+1)=9, then k^2+p^2 is divisible by 5.​

Is that right?

Shall we take that to imply that a is positive, and that k and p must be integers?

I find that I can solve for specific values of k and p, which makes the problem feel like we're given too much.
 
I worked it out assuming that a>0. I made no assumption that p and k were integers. It turned out that they were but in theory they didn't have to. Just k2+p2 had to be an integer as that would be a necessary condition for 5 to divide k2+p2
 
Helenam said:
Not sure how to start this question.
If f(k/2)=1 and f(p+1)=9. Show that k^2+p^2 is divisible by 5.
Any help would be appreciated. Thanks.
Click to expand...
What is the domain of each of those numbers?

Can k & p & a be fractions? Can those be negative numbers ?

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

READ BEFORE POSTING

Please share your work/thoughts about this problem
 
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