Let's analyze each given function to determine which one is a polynomial function.
### Definition of a Polynomial Function
A polynomial function is a mathematical expression that represents a sum of terms, each consisting of a variable raised to a non-negative integer exponent and multiplied by a coefficient. The general form of a polynomial function is:
[tex]\[ f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \][/tex]
where [tex]\(a_n, a_{n-1}, \ldots, a_1, a_0\)[/tex] are constants and [tex]\(n\)[/tex] is a non-negative integer.
### Analyzing Each Function
1. [tex]\( f(x) = 5x^3 + 2x^2 - 6x + 7 \)[/tex]
- This function is a sum of terms where the variable [tex]\(x\)[/tex] is raised to the non-negative integer powers 3, 2, 1, and 0.
- The coefficients are 5, 2, -6, and 7, respectively.
- This fits the definition of a polynomial function.
2. [tex]\( g(x) = \frac{x^2 + 2x - 3}{x - 4} \)[/tex]
- This function involves division by [tex]\((x - 4)\)[/tex].
- The presence of a variable in the denominator means it is not a polynomial function.
3. [tex]\( h(x) = 750(1.04)^x \)[/tex]
- This is an exponential function because the variable [tex]\(x\)[/tex] is in the exponent.
- Exponential functions are not polynomial functions.
4. [tex]\( p(x) = \log_3(x)^2 \)[/tex]
- This function involves the natural logarithm of [tex]\(x\)[/tex] raised to a power.
- Logarithmic functions are not polynomial functions.
### Conclusion
Based on the analysis, the function that fits the definition of a polynomial function is:
[tex]\[ f(x) = 5x^3 + 2x^2 - 6x + 7 \][/tex]
Thus, the polynomial function among the given options is:
[tex]\[ f(x) = 5x^3 + 2x^2 - 6x + 7 \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{f(x) = 5x^3 + 2x^2 - 6x + 7} \][/tex]
