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Chapter 3: Problem 10

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

Short Answer

Expert verified
The function \(f(x)=\frac{x^{2}+7}{3}\) is a polynomial function and its degree is 2.

Step by step solution

01

Identify Variables and Structure

In the function \(f(x)=\frac{x^{2}+7}{3}\), \(x\) is the variable and the function involves \(x^{2}\), a variable to the power of 2, as well as a constant (7), divided by another constant (3).
02

Determine if the Function Corresponds to a Polynomial

Polynomials can have terms with a variable raised to a non-negative integer power, multiplied by coefficients. The function \(f(x)=\frac{x^{2}+7}{3}\) indeed fulfils that condition, because \(x^{2}\) has a non-negative integer exponent (2), and both \(x^{2}\) and the constant (7) are multiplied by coefficients (1 and 1/3 respectively). Therefore, \(f(x)=\frac{x^{2}+7}{3}\) is indeed a polynomial function.
03

Identify the Degree of the Polynomial

The degree of the polynomial is the highest power of the variable in the polynomial. In the function \(f(x)=\frac{x^{2}+7}{3}\), the highest power of \(x\) is 2, therefore the degree of the polynomial is 2.

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