Let's analyze the given function $g(x) = x^{7/2} - x^4 + 9$.

Step 1: Determine if the function is a polynomial

A polynomial function has the general form:

$P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$

where:

• The exponents of $x$ must be nonnegative integers (i.e., 0, 1, 2, 3,...).
• The coefficients $a_n, a_{n-1}, \dots, a_0$ are real numbers.

Analyzing each term:

• $x^{7/2}$ has an exponent of $7/2$, which is not a nonnegative integer.
• $-x^4$ has an exponent of 4, which is a nonnegative integer.
• The constant term $9$ can be viewed as $9x^0$, and the exponent 0 is also a nonnegative integer.

Because the first term $x^{7/2}$ has an exponent that is not a nonnegative integer, this disqualifies the function from being a polynomial.

Step 2: Conclusion

Since the variable $x$ is raised to the power of $7/2$ (which is not a nonnegative integer), this function is not a polynomial.

Thus, the correct answer is:

B. It is not a polynomial because the variable $x$ is raised to the $7/2$ power, which is not a nonnegative integer.

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Would you like further clarification on this topic? Here are some related questions to help you deepen your understanding:

1. What are the basic properties of a polynomial function?
2. Can a fractional exponent ever result in a polynomial?
3. How do you find the degree of a polynomial?
4. What distinguishes a monomial from a polynomial?
5. How do you identify the leading term of a polynomial?

Tip: A polynomial can have only whole-number exponents (including zero). Any fractional or negative exponent will disqualify it from being a polynomial.