Determine which functions are polynomial functions. For those that are, identify the degree. $f\left(x\right)=x^{\frac{1}{2}}-3x^2+5$

Verified step by step guidance

1. Recall that a polynomial function is a function of the form \(f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0\), where each exponent \(n\) is a non-negative integer (0, 1, 2, ...), and the coefficients \(a_i\) are real numbers.
2. Examine the given function: \(f(x) = x^{1/2} - 3x^2 + 5\). Identify the exponents of each term: the first term has exponent \(\frac{1}{2}\), the second term has exponent 2, and the last term is a constant (exponent 0).
3. Check if all exponents are whole numbers (non-negative integers). Since \(\frac{1}{2}\) is not an integer, the term \(x^{1/2}\) is not allowed in a polynomial function.
4. Conclude that because of the \(x^{1/2}\) term, the function \(f(x)\) is not a polynomial function.
5. If the function were a polynomial, the degree would be the highest exponent among the terms. Here, ignoring the non-polynomial term, the highest integer exponent is 2, so the degree would be 2 if it were a polynomial.

Key Concepts

Polynomial Functions

A polynomial function is a function that can be expressed as a sum of terms consisting of variables raised to non-negative integer powers multiplied by coefficients. For example, f(x) = 2x^3 - 4x + 7 is a polynomial, but functions with variables under roots or with negative or fractional exponents are not.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable with a non-zero coefficient. It indicates the polynomial's order and affects its graph's shape and behavior. For instance, in f(x) = 4x^5 + 2x^3, the degree is 5.

Identifying Non-Polynomial Terms

Terms with variables raised to fractional or negative exponents, or variables inside roots, are not part of polynomial functions. For example, x^(1/2) (square root of x) is not a polynomial term because the exponent is a fraction, violating the non-negative integer exponent rule.