# Are the Sum of Two Functions Always Equal to the Sum of their Individual Parts?

• coverband
In summary, the notation (f+g)(x) represents the composition of two functions, f(x) and g(x), where the output of g(x) is used as the input for f(x). To prove (f+g)(x) = f(x) + g(x), you must show that they have the same domain and range, and that they produce the same output for every input. Whether (f+g)(x) can be simplified further depends on the specific functions f(x) and g(x). It is commutative, and it is different from the traditional notation for function composition.
coverband
where f and g are finctions of x

you mean "functions"...

that's a definition as far as I am concerned. How are you defining (f+g)(x)?

This appears to be the definition of the sum of two functions. It is valid as long as f and g have the same domains and ranges.

ObsessiveMathsFreak said:
This appears to be the definition of the sum of two functions. It is valid as long as f and g have the same domains and ranges.

They don't need the same range.

Try sin(x) + x.

## 1. What is the definition of (f+g)(x)?

The notation (f+g)(x) represents the composition of two functions, f(x) and g(x). It means that the output of g(x) is used as the input for f(x), or in other words, g(x) is "plugged in" to f(x).

## 2. How do you prove (f+g)(x) = f(x) + g(x)?

To prove that two functions are equal, you must show that they have the same domain and range, and that they produce the same output for every input. In the case of (f+g)(x) = f(x) + g(x), you would need to show that for any given value of x, both sides of the equation produce the same result.

## 3. Can (f+g)(x) be simplified further?

It depends on the specific functions f(x) and g(x). Some combinations of functions can be simplified, while others cannot. For example, if f(x) = 2x and g(x) = x^2, then (f+g)(x) = 2x + x^2, which cannot be simplified further. Other combinations, such as f(x) = 2x and g(x) = -2x, can be simplified to (f+g)(x) = 0.

## 4. Is (f+g)(x) commutative?

Yes, (f+g)(x) is commutative, meaning that the order in which the functions are composed does not affect the result. This can be seen in the equation (f+g)(x) = f(x) + g(x), where the order of f(x) and g(x) can be switched without changing the outcome.

## 5. How is (f+g)(x) related to the composition of functions?

As mentioned in the answer to the first question, (f+g)(x) is a composition of two functions, f(x) and g(x). However, (f+g)(x) is different from the traditional notation for function composition, which is written as f(g(x)). In this case, (f+g)(x) is a combination of two functions, while f(g(x)) is a function that takes the output of g(x) as the input for f(x).

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