Respuesta :
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.
The set can not be a vector space.
Vector space :
It is a set of objects called vectors, which may be added together and multiplied by numbers called scalars.
- If function [tex]f(x)[/tex] and [tex]g(x)[/tex] are elements of the set,
- then the sum [tex]f(x)+g(x)[/tex] should be an element of the set.
- But the sum of quadratic functions may not compulsory be a quadratic function.
Let us consider that, [tex]f(x) = x^{2} + x + 5[/tex] and [tex]g(x) = -x^{2} +5[/tex]
It is observed that Both f and g are quadratic functions and the graphs of both functions contain the point (0,5)
However [tex]f(x) + g(x) = (x^{2} +x+5)+(-x^{2} +5) = x+10[/tex]
which is not a quadratic function.
And [tex]f(x)+g(x)[/tex] not passes through [tex](0,5)[/tex]
It shows that [tex]f+g[/tex] is not in the set.
Thus, , the set can not be a vector space.
Learn more about the vector space here:
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