A polynomial function is a function which is a polynomial, which means that the variables' exponents are positive integers and their coefficients are real numbers. Let's examine one function at a time.
We see that the function x5+5x4−x2−7 has variable terms whose exponents are positive integers and coefficients are real, which means it is a polynomial. Therefore, the function p(x) is a polynomial function.
The expression x1 can be rewritten as x-1. We see that the exponent is not a positive integer, which means q(x) is not a polynomial function.
Here, we rewrite the function expression in order to see if it's a polynomial. A square root is the same thing as raising to the power of 21, which means the function expression can be written as x1/2+2. We see that the exponent is not a positive integer, which means f(x) is not a polynomial function.
The third term in the expression x7+x6+x-5−1 has a negative exponent. Thus, it is not a positive integer. The function g(x) is not a polynomial function.
The function expression 4x2+x−3 satisfies the conditions of having positive integer exponents and real coefficients, which means h(x) is a polynomial function.