Let us determine the closure of polynomial functions under subtraction by examining the forms of two polynomials, [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex].
Given:
[tex]\[
f(x) = a_0 + a_1 x + a_2 x^2 + \ldots + a_n x^n
\][/tex]
[tex]\[
g(x) = b_0 + b_1 x + b_2 x^2 + \ldots + b_m x^m
\][/tex]
To understand the subtraction of two polynomials, we subtract [tex]\( g(x) \)[/tex] from [tex]\( f(x) \)[/tex]:
[tex]\[
h(x) = f(x) - g(x)
\][/tex]
Substitute the expressions for [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] into the equation:
[tex]\[
h(x) = (a_0 + a_1 x + a_2 x^2 + \ldots + a_n x^n) - (b_0 + b_1 x + b_2 x^2 + \ldots + b_m x^m)
\][/tex]
Separate and group the like terms:
[tex]\[
h(x) = (a_0 - b_0) + (a_1 - b_1)x + (a_2 - b_2)x^2 + \ldots + (a_k - b_k)x^k
\][/tex]
Here, [tex]\( k \)[/tex] is the maximum of [tex]\( n \)[/tex] and [tex]\( m \)[/tex], since some polynomials may have different degrees and a polynomial of a lower degree can be thought to have zero coefficients for the higher-degree terms it does not actually have.
Each term in the polynomial [tex]\( h(x) \)[/tex] is of the form [tex]\( (a_i - b_i)x^i \)[/tex], where [tex]\( (a_i - b_i) \)[/tex] is a real number (because the coefficients of polynomials are real numbers subtracted from each other). The resulting polynomial [tex]\( h(x) \)[/tex] is thus:
[tex]\[
h(x) = c_0 + c_1 x + c_2 x^2 + \ldots + c_k x^k
\][/tex]
where [tex]\( c_i = a_i - b_i \)[/tex] for the common degrees of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex], and either [tex]\( a_i \)[/tex] or [tex]\( b_i \)[/tex] for degrees that only one polynomial reaches if the other does not.
Therefore, the subtraction of two polynomials is itself a polynomial. This demonstrates that the set of polynomial functions is closed under subtraction.
The correct answer is:
[tex]\[
\boxed{\text{C.} \; f(x) \; \text{and} \; g(x) \; \text{are closed under subtraction because when subtracted, the result will be a polynomial.}}
\][/tex]
