Question

Determine whether the set, together with the indicated operations, is a vector space. If it is not, then identify one of the vector space axioms that fails.
(a) The set of all quadratic functions whose graphs pass through the point (0, 5) with the standard operations.

Answer 1

Answer:

It isn't a vector space

Step-by-step explanation:

If f and g were elements of the set, then the sum f+g should be an element of the set. However the sum of quadratic funcitons may not neccesarily be a quadratic function. Lets look at

f(x) = x²+ x + 5

g(x) = -x²+5

Both f and g are quadratic functions and the graphs of both functions contain the point (0,5), because f(0) = g(0) = 5. However f(x) + g(x) = (x²+x+5)+(-x²+5) = x+10, which is not a quadratic funciton. Furthermore, f+g(0) = 10, so its graph doesnt contain the point (0,5). This shows that f+g is not in the set, therefore, the set cant be a vector space.