Undergrad The sum of these functions equals a constant

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The discussion centers on whether the equation f(x) + g(x) = c, where c is a constant, implies that both f(x) and g(x) must also be constants. It is clarified that this is not necessarily true, as demonstrated by the example of f(x) = cos(x) and g(x) = -cos(x) + 1, which sum to a constant but are not constant functions themselves. The conversation emphasizes the distinction between equations that hold for specific values of x versus those that are identities true for all x. The conclusion drawn is that while the sum can equal a constant, it does not require the individual functions to be constant. Therefore, the functions can vary while still satisfying the equation.
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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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?
 
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.
 
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).
 
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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