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.
You must understand first what is a function ! Suppose f(x) is a function of x, then what is the meaning of it ! It means, you take a real number "x" and put it in front of "f" then "f" applies a rule and gives another real number, in that sense f(x) is again a real number, mathematically:
f : x ---> y, x,y belongs to real numbers.
as for example f(x) = x^2, here "f" is a rule that "squares" a real number "x" when given to it:
f: x ----- squares-----> y = x^2
so for x = 2, f(x) = 4.
So f(x) is a number also, so all the operations those apply to a real number also apply to a function of a real number.