How can I determine this function?

Question

If i know that$$ f(3)-f(-1/2) = 7$$ and $$ f(2)-f(1/2) = 3 $$ and $$f(3/2)-f(-2) =7$$ How can I determine the function?

Even if these equations allowed you to determine $f(x)$ at these six locations (they can't), that is still only six locations out of infinitely many; so there is not nearly enough information. — vadim123, Oct 31 '13 at 18:17
You can’t, without more information. Judging by those numbers, though, I suspect that $f$ is supposed to be a linear function, in which case you can at least pin down some information about $f$; is that the case? — Brian M. Scott, Oct 31 '13 at 18:18
You need more information (such as smoothness and others) about $f$ to determine it by the above equations. — Lei Li, Oct 31 '13 at 18:21
Your numbers are consistent with $f'(x)=\frac12$, but also with many other more complicated expressions — Henry, Oct 31 '13 at 18:22
Where'd you get this problem? There' not enough info to determine the function. — Stefan Smith, Nov 01 '13 at 01:26

score 1 · Answer 1 · answered Oct 31 '13 at 19:04

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$$ f(x) = \left\{ \begin{array}{ll} 110004 & \quad x = -2 \\ 68 & \quad x = -\frac{1}{2} \\ 4 & \quad x = \frac{1}{2} \\ 110011 & \quad x = \frac{3}{2} \\ 7 & \quad x = 2 \\ 75 & \quad x = 3 \\ 0 & \quad x\neq-2,-\frac{1}{2},\frac{1}{2},\frac{3}{2},2,3 \end{array} \right. $$

answered Oct 31 '13 at 19:04

Rocket Man

2,433

Could you explain this?, I don't get it. – Satish Ramanathan Oct 31 '13 at 20:09
Look at the given information by Veritas. My function satisfies those criteria. – Rocket Man Oct 31 '13 at 20:26
Got it!! You could have multiple such combinations – Satish Ramanathan Oct 31 '13 at 20:29

J. W. Perry · Answer 2 · 2013-10-31T18:57:22.800

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One such function would be any linear function of the form $$f(x)=2x+c.$$ This comes from observing that the average rate of change (think slope) on each interval is constant, $$\frac{f(b_i)-f(a_i)}{b_i-a_i}=\frac{7}{3-\left(-\frac{1}{2}\right)}=\frac{3}{2-\left(\frac{1}{2}\right)}=\frac{7}{\frac{3}{2}-(-2)}=2.$$ My guess is that you are being prodded to answer $f(x)=2x+c,\,c\in\mathbb{R}$.

edited Oct 31 '13 at 18:57

answered Oct 31 '13 at 18:34

J. W. Perry

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score 0 · Answer 3 · answered Oct 31 '13 at 18:35

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Seems to be that f(x) = 2x could satisfy this but I wonder there are a lot more functions that can satisfy this if we work out!!

answered Oct 31 '13 at 18:35

Satish Ramanathan

11,984

score 0 · Answer 4 · answered Oct 31 '13 at 18:40

$$f(3) - f(-1/2) = f(3/2)-f(2)$$ this gets you $f(7/2) = 7 $ and you also know that $f(3/2) = 3$. And by substituting $f(3/2) = 3 $ you can also get $f(-2) = -4$I agree with the others that there is information missing and I made some assumptions about the function at hand, but a possible answer is $f(x) = 2x$

How can I determine this function?

4 Answers4