Sketching Polynomial Functions

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.

$p(x)$

We see that the function $x^5+5x^4-x^2-7$ 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.

$q(x)$

The expression $\frac{1}{x}$ can be rewritten as $x^{\text{-}1}.$ We see that the exponent is not a positive integer, which means $q(x)$ is not a polynomial function.

$f(x)$

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 $\frac{1}{2},$ which means the function expression can be written as $x^{1/2}+2.$ We see that the exponent is not a positive integer, which means $f(x)$ is not a polynomial function.

$g(x)$

The third term in the expression $x^7+x^6+x^{\text{-}5}-1$ has a negative exponent. Thus, it is not a positive integer. The function $g(x)$ is not a polynomial function.

$h(x)$

The function expression $4x^2+x-3$ satisfies the conditions of having positive integer exponents and real coefficients, which means $h(x)$ is a polynomial function.