What Functions Satisfy f(x)f(-x) = 1?

  • Context: Undergrad 
  • Thread starter Thread starter Mr Davis 97
  • Start date Start date
  • Tags Tags
    Property
Click For Summary

Discussion Overview

The discussion revolves around identifying functions that satisfy the equation ##f(x)f(-x) = 1##. Participants explore various types of functions, including continuous and piecewise continuous examples, and consider the implications of differentiability and continuity in their solutions.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • Some participants propose that exponential functions, such as ##f(x) = e^x##, satisfy the equation.
  • Others suggest that if ##f(0) = \pm 1##, then one can freely choose ##f(x)## for ##x > 0## as long as it does not equal zero.
  • One participant mentions that if ##f(x)## is continuous, functions like ##f(x) = \exp(x^n)##, where ##n## is an odd integer, are solutions.
  • Another example provided is a piecewise function defined differently for different intervals.
  • Some participants discuss the implications of differentiability, suggesting that if ##f(x)## is continuous and differentiable for ##x > 0##, it may be differentiable everywhere.
  • Several participants mention constant functions, such as ##f(x) = 1##, as trivial solutions.
  • One participant introduces the function ##f(x) = |x|^x##, assuming a value of ##f(x) = 1## at ##x = 0## where it is otherwise undefined.

Areas of Agreement / Disagreement

Participants express a variety of viewpoints regarding the types of functions that satisfy the equation, with no clear consensus on a single solution or method. Some agree on the validity of certain functions, while others challenge or refine these claims.

Contextual Notes

Limitations include the dependence on the continuity and differentiability of the functions discussed, as well as the assumptions made about the values at specific points, such as ##f(0)##.

Mr Davis 97
Messages
1,461
Reaction score
44
Out of curiosity, I'm trying to find functions of a real variables such that ##f(x)f(-x) = 1##. One obvious example is ##f(x) = e^x##, and all other exponential functions. Are there any other examples? How would I go about generating them?
 
Mathematics news on Phys.org
You already did. ##f(0)=\pm 1## and there is a differential equation
 
Freely choose f(x) for x>0 as long as no value is zero. Choose -1 or 1 for f(0). Fix f(x) for negative x according to your equation.
 
If ##f(x)## has to be continuous, the functions ##f(x) = \exp (x^n )##, where ##n## is an odd integer, are one example of a solution.

Piecewise continuous examples include functions like

##f(x) = \left\{\begin{array}{c} 2^x, \hspace{30pt} |x|<5\\3^x, \hspace{30pt}|x|\geq 5\end{array}\right.##
 
Ugly, but correct. I was looking for too decent :rolleyes: functions and thought to exploit ##f'(x) f(-x) + f(x) f'(-x) = 0 ## which is satisfied if ##f'(x) = \pm f(x)\ \forall x\ ## and also if ##f'(x) = 0\ \forall x\ ## . But compared to mfb, that's far too limited
 
If f(x) has to be continuous then we have the options f(0)=1 and f(x)>0 arbitrary but continuous for x>0, or f(0)=-1 and f(x)<0 arbitrary but continuous for x>0. In both cases we can again find the negative function values with the given constraint and it will always be continuous there as well.

If f(x) for x>0 is differentiable and the limit of the derivative for x->0 is finite then I would expect f(x) to be differentiable everywhere.

This works both for real and complex function values.
 
  • Like
Likes   Reactions: BvU
f(x)=x^{2n}, f(x)=\lvert x \rvert, f(x)= \cos(\pi x),...
 
Huh ?
 
  • #10
mfb said:
You might want to check that @Svein.
Sorry, did not read the specs closely enough (read it as f(1)f(-1)=1).

But then the constant function f(x)=1 is a trivial solution.
 
Last edited:
  • #11
Svein said:
But then the constant function f(x)=1 is a trivial solution.
Right. It is a special case of ##f(x)=k^x## which was mentioned in post #1 ("all exponential functions").
 
  • #12
Even the function ##f(x) = |x|^x## is a solution if value ##f(x) = 1## is assumed at ##x=0## where it is undefined.
 
  • #13
jbriggs444 said:
Right. It is a special case of ##f(x)=k^x## which was mentioned in post #1 ("all exponential functions").
Another trivial solution is f(x)=-1...
 

Similar threads

Undergrad Comparing ##a^b## and ##b^a##
  • · Replies 19 ·
Replies
19
Views
1K
Undergrad Can the same argument be used for both radians and degrees in the sine function?
  • · Replies 3 ·
Replies
3
Views
2K
MHB Is $f(x) = xf(1)$ the only solution to the given functional equation?
  • · Replies 3 ·
Replies
3
Views
2K
High School Find the function f(x) satisfy: x=0,-> f(x) =0, x<>0, f(x)=1
  • · Replies 10 ·
Replies
10
Views
3K
MHB Finding $a$ and $b$ for $f(x)=|\lg (x+1)|$
  • · Replies 3 ·
Replies
3
Views
1K
Undergrad Best way to fit three functions
  • · Replies 2 ·
Replies
2
Views
1K
Undergrad Transformation of Functions: How Do Domain and Range Change?
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad What is the function that describes this Asymptotic behaviour?
  • · Replies 13 ·
Replies
13
Views
2K
Find f s. t. ||f||=1 and f(x) < 1 with ||x||=1
  • · Replies 20 ·
Replies
20
Views
3K
Undergrad ##f(2x)=f^2(x)-2f(x)-1/2## then find ##f(3)##
  • · Replies 17 ·
Replies
17
Views
4K