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How can you define the sum and product of two functions

Let f and g be functions from a set A to the set of real numbers R.

Then the sum and the product of f and g are defined as follows:

For all x, ( f + g )(x) = f(x) + g(x) , and

for all x, ( f*g )(x) = f(x)*g(x) ,

where f(x)*g(x) is the product of two real numbers f(x) and g(x).

Example: Let f(x) = 3x + 1 and g(x) = x2 . Then ( f + g )(x) = x2 + 3x + 1 , and ( f*g )(x) = 3x3 + x2