Explain how operations with functions are similar to operations with real numbers.


Considering as example the single variable function, you should remember that a function uniquely relates real values such that the result is a set of ordered pairs `(x,f(x)).`

The relation between inputs and outputs is given by an equation called the equation of function.

Since the outputs are also real values, hence, reasoning by analogy, you may use the arithmetical operation between functions (addition, subtraction, multiplication, division) to produce new functions.

Addition of two functions f(x) and g(x) is defined `f(x)+g(x) = (f+g)(x)` .

Subtraction of two functions f(x) and g(x) is defined `f(x)-g(x) = (f-g)(x).`

Multiplication of two functions f(x) and g(x) is defined `f(x)*g(x) = (f*g)(x)` .

Division of two functions f(x) and g(x) is defined `f(x)/g(x) = (f/g)(x).`

Considering as example f(x) = x-1, g(x) = x+1, you may use the arithmetical operations in case of these functions such that:

`f(x)+g(x) = x-1+x+1 = 2x`

Notice that giving values to x, the result is also a value.

`f(x)-g(x) = x-1-x-1 = -2`

`f(x)*g(x) = (x-1)(x+1) = x^2-1`

`f(x)/(g(x)) = (x-1)/(x+1)`

Hence, since the functions operate with real numbers then the arithmetical operation hold for functions.

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