Chapter 4, Problem 3RE

In Problems $1-4$, determine whether the function is a polynomial function, a rational function, or neither. For those that are polynomial functions, state the degree. For those that are not polynomial functions, tell why not.

$f\left(x\right)=3x^2+5x^{\frac{1}{2}}-1$

To determine

Whether the function $f(x)=3x^2+5x^{\frac{1}{2}}-1$ is a polynomial function, a rational function, or neither. If polynomial, state degree, otherwise tell why not.

Answer to Problem 3RE

Solution:

The function $f(x)=3x^2+5x^{\frac{1}{2}}-1$ is neither polynomial nor rational.

It is not polynomial function because in the second term $5x^{\frac{1}{2}}$ the variable $x$ is raised to the $\frac{1}{2}$ power, which is not a nonnegative integer.

Explanation of Solution

Given Information:

The function, $f(x)=3x^2+5x^{\frac{1}{2}}-1$

The function $f(x)$ is not polynomial function because in the second term $5x^{\frac{1}{2}}$ the variable $x$ is raised to the $\frac{1}{2}$ power, which is not a nonnegative integer.

The function $f(x)$ is not a rational function because it is not denoted in the form of ratio of polynomial functions.

Hence, the function $f(x)=3x^2+5x^{\frac{1}{2}}-1$ is neither polynomial nor rational.