The sum of these functions equals a constant

In summary: This means that the two functions are related, but not necessarily equal. It is possible for both functions to be non-zero but still produce a sum of zero. So no, the functions being equal to zero is not a necessary condition for the sum to equal zero.In summary, the conversation discusses whether the sum of two functions being a constant implies that both functions are constant. The conclusion is that it does not, and an example is provided to illustrate this. It is also mentioned that the functions being equal to zero is not a necessary condition for the sum to equal zero.
  • #1
kent davidge
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If I have a sum ##f(x) + g(x) = c##, with ##c## a constant, does this imply that both ##f(x)## and ##g(x)## are also constants?
If I just solve this equation for ##x##, I will find some values of ##x## which satisfy the equation. However, if I require that the equation be true for all ##x##, there is no way other than the functions being equal to constants. Am I right?
 
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  • #2
kent davidge said:
If I have a sum ##f(x) + g(x) = c##, with ##c## a constant, does this imply that both ##f(x)## and ##g(x)## are also constants?
No, not at all. Consider ##f(x) = \cos x## and ##g(x) = -\cos x + 1##. Neither function is constant, but their sum is always equal to 1.
kent davidge said:
If I just solve this equation for ##x##, I will find some values of ##x## which satisfy the equation. However, if I require that the equation be true for all ##x##, there is no way other than the functions being equal to constants. Am I right?
No. With my example functions, you can't solve the equation ##f(x) + g(x) = 1## for x. My equation is true for all real x.

Also, it depends on whether the equation you're working with is true only for a certain number of values of x, or is an identity, one that is true for all values in the domains of the two functions.

Can you be more specific about what you're trying to do?
 
  • #3
Mark44 said:
Can you be more specific about what you're trying to do?
I just have two seemgly arbritary functions of ##x## whose sum is zero, and wanted to know if that implies that the functions themselves are equal to zero, but now with your answer I see that it doesn't.
 
  • #4
kent davidge said:
If I just solve this equation for x, I will find some values of x which satisfy the equation.
All x should satisfy the equation if that equation is a general relation ("c is a constant" implies that).
 
  • #5
kent davidge said:
I just have two seemgly arbritary functions of ##x## whose sum is zero, and wanted to know if that implies that the functions themselves are equal to zero, but now with your answer I see that it doesn't.
If ##f(x) + g(x) = 0## then ##g(x) = -f(x)##.
 
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Related to The sum of these functions equals a constant

1. What does it mean when the sum of these functions equals a constant?

When the sum of these functions equals a constant, it means that the output of the functions when added together will always result in the same value, regardless of the input.

2. Can the sum of two functions equal a constant if the functions are not related?

No, the sum of two functions can only equal a constant if the functions are related in some way. This could be through a mathematical relationship, such as being inverses of each other, or through a physical relationship, such as representing different aspects of the same system.

3. How can I determine the value of the constant when given two functions?

To determine the value of the constant when given two functions, set the two functions equal to each other and solve for the constant. This will give you the value that the sum of the functions will always equal.

4. Can the sum of functions equal a constant if one or more of the functions is a constant itself?

Yes, the sum of functions can equal a constant if one or more of the functions is a constant itself. In fact, this is a common occurrence in mathematical equations and can be useful in solving for unknown variables.

5. Is the sum of functions equaling a constant a common concept in science?

Yes, the concept of the sum of functions equaling a constant is common in science. It is often used in mathematical modeling and can be seen in various fields such as physics, chemistry, and biology. It is a useful tool for understanding and predicting the behavior of systems and processes.

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