To determine whether [tex]\( f(x) = 2 - \frac{1}{x^3} \)[/tex] is a polynomial function or not, we need to check the general form and properties of polynomial functions.
A polynomial function is defined by the following general form:
[tex]\[ f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \][/tex]
where [tex]\( a_n, a_{n-1}, \ldots, a_0 \)[/tex] are constant coefficients and [tex]\( n \)[/tex] is a nonnegative integer.
Key characteristics of polynomial functions include:
1. The variables must only have nonnegative integer exponents.
2. There must be no variables in the denominators.
3. There must be no negative or fractional exponents.
Let’s examine the given function [tex]\( f(x) = 2 - \frac{1}{x^3} \)[/tex]:
1. The term [tex]\( 2 \)[/tex] is a constant, which is acceptable in a polynomial.
2. The term [tex]\( -\frac{1}{x^3} \)[/tex] can be rewritten using a negative exponent as [tex]\( -x^{-3} \)[/tex].
In this rewritten form, [tex]\( -x^{-3} \)[/tex] indicates that the variable [tex]\( x \)[/tex] is raised to a negative power ([tex]\( -3 \)[/tex]).
Since polynomial terms must have variables raised to nonnegative integer exponents, the term [tex]\( -x^{-3} \)[/tex] violates this requirement.
Therefore, [tex]\( f(x) = 2 - \frac{1}{x^3} \)[/tex] is not a polynomial function.
The correct choice is:
B. It is not a polynomial because the variable [tex]\( x \)[/tex] is raised to the [tex]\(-3\)[/tex] power, which is not a nonnegative integer.
