The set of all real-valued functions f defined everywhere on the real line and such that f(4) = 0, with the operations (f+g)(x)=f(x)+g(x) and (kf)(x)=kf(x)
verify if all axioms hold true.
Axioms 1 and 6 These closure axioms require that if we add two functions that are defined at each x in the interval , then sums and scalar multiples of those functions are also defined at each x in the interval.
Axiom 4 This axiom requires that there exists a function 0 in , which when added to any other function f in produces f back again as the result. The function, whose value at every point x in the interval is zero, has this property. Geometrically, the graph of the function 0 is the line that coincides with the x-axis.
Axiom 5 This axiom requires that for each function fin there exists a function —f in , which when added to f produces the function 0. The function defined by has this property. The graph of can be obtained by reflecting the graph of f about the x-axis
Axioms 2,3,7,8,9,10 The validity of each of these axioms follows from properties of real numbers. For example, if f and g are functions in , then Axiom 2 requires that . This follows from the computation
The Attempt at a Solution
I couldn't find a good example in my textbook so I kinda don't know how to start this problem. Help is much appreciated.