Subtraction of real function from another

[Abu Rehan]

We define subtraction of real functions as (f-g)(x) = f(x) - f(x). But I wonder what does (f-g) mean! I don't find any meaning in (sin - (square of))(x).
If it really has any meaning then tell with examples.

 

[snipez90]

h = f -g denotes the actual function which takes an element x in the domain (here it's the intersection of the domains of f and g) and outputs f(x) - g(x). This is why we write (f-g)(x) = f(x) - g(x). Remember f and g are functions, not numbers, whereas f(x) and g(x) are real numbers, if f and g are real valued. While it's important to keep in mind the distinction between a function f and the values of its output f(x) (x in the domain of f), it's rather common to write f(x) to denote the function (e.g. among mathematical analysts), usually for convenience.

 

[pellman]

We define subtraction of real functions as (f-g)(x) = f(x) - f(x). But I wonder what does (f-g) mean! I don't find any meaning in (sin - (square of))(x).
If it really has any meaning then tell with examples.

In the example you give (sin - (square of))(x) would equal sin(x) + x^2. It's really just that simple.

 

[HallsofIvy]

We define subtraction of real functions as (f-g)(x) = f(x) - f(x). But I wonder what does (f-g) mean! I don't find any meaning in (sin - (square of))(x).
If it really has any meaning then tell with examples.

The whole purpose of "(f- g)(x)= f(x)- g(x)" is to tell you what f- g means.

(sin- square of)(x) is, according to that definition, $sin(x)- square of (x)= sin(x)- x^2$

 

[CellCoree]

Subtraction of real functions is a common operation in mathematics and is used to find the difference between two functions. The notation (f-g)(x) simply means that we are subtracting the function g(x) from the function f(x). In other words, we are finding the difference between the outputs of the two functions at a specific input value x.

To better understand this concept, let's consider an example. Let f(x) = x and g(x) = 2x. Now, (f-g)(x) = f(x) - g(x) = x - 2x = -x. This means that at any given input value x, the output of (f-g)(x) is equal to the output of f(x) minus the output of g(x).

In your specific example of (sin - (square of))(x), we can rewrite it as (sin(x) - (x^2)) and understand it as the difference between the sine function and the square function at a specific input value x. For instance, if we take x = 2, then (sin - (square of))(2) = sin(2) - (2^2) = 0.9093 - 4 = -3.0907.

In general, the subtraction of real functions is a useful tool in mathematical analysis and is often used to find the difference between two quantities or to solve equations. I hope this explanation helps clarify the meaning and purpose of (f-g) in the context of real functions.