To determine which of the given functions are polynomial functions, let's analyze each one:

1. $g(x) = 3 - x^4 + 2x^3$:

   • This is a polynomial function because it is a sum of terms with non-negative integer exponents (0, 3, and 4).

2. $f(x) = x^2 - 3x + 5$:

   • This is also a polynomial function, as it contains terms with non-negative integer exponents (0, 1, and 2).

3. $m(x) = x^{-2} + 3x^2$:

   • This is not a polynomial function because it includes the term $x^{-2}$, which has a negative exponent.

4. $h(x) = \sqrt{x} - x^{1/3} - 5x$:

   • This is not a polynomial function because it contains terms with fractional exponents ($\sqrt{x} = x^{1/2}$ and $x^{1/3}$).

Conclusion

The functions that are polynomial functions are:

• $g(x) = 3 - x^4 + 2x^3$
• $f(x) = x^2 - 3x + 5$

Do you have any questions or would you like to discuss this further?

Related Questions:

1. What are the properties of polynomial functions?
2. How do you determine the degree of a polynomial?
3. What distinguishes a polynomial function from a rational function?
4. Can a polynomial function have negative exponents?
5. How does the graph of a polynomial function differ from non-polynomial functions?

Tip: Polynomial functions cannot have variables in the denominator or with negative or fractional exponents.