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Chapter 3: Problem 8

Determine which functions are polynomial functions. For those that are, identify the degree. $$f(x)=x^{\frac{1}{3}}-4 x^{2}+7$$

Short Answer

Expert verified
The function \(f(x)=x^{1/3} -4x^{2}+7\) is not a polynomial function.

Step by step solution

01

Identify The Powers of x

Here in \(f(x)=x^{1/3} -4x^{2}+7\), we notice that there are three terms. The first one is \(x^{1/3}\), the second one is \(x^{2}\) and the third one is \(7=x^{0}\). Notice that the powers of x are \(1/3\), \(2\), and \(0\) respectively.
02

Check If All Powers Are Nonnegative Integers

It's important to check if all powers of x are nonnegative integers. This must be the case if the function is a polynomial. Here, in the first term, the power of x is \(1/3\), which is not an integer. Therefore, this function is not a polynomial function because it doesn't meet the criteria.
03

Determine The Degree

Since it's established that the function is not a polynomial function, the step to determine the degree is not necessary. However, if the function was a polynomial, then the degree would have been equal to the highest power of x present in the function.

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Most popular questions from this chapter

What does it mean if two quantities vary directly?

Write an equation that expresses each relationship. Then solve the equation for \(y .\) \(x\) varies jointly as \(y\) and \(z\) and inversely as the square of \(w\).

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Use a graphing utility to graph \(y=\frac{1}{x}, y=\frac{1}{x^{3}},\) and \(\frac{1}{x^{5}}\) in the same vicwing rectangle. For odd values of \(n\), how does changing \(n\) affect the graph of \(y=\frac{1}{x^{m}} ?\)

Solve each inequality using a graphing utility. $$ x^{3}+x^{2}-4 x-4>0 $$
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