0. Functions

Common Functions

0. Functions

Common Functions

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Multiple Choice

Determine if the given function is a polynomial function. If so, write in standard form, then state the degree and leading coefficient. $f\left(x\right)=4x^3+\frac12x^{-1}-2x+1$

Polynomial with $n=3,a_{n}=4$

Polynomial with $n=4,a_{n}=3$

Polynomial with $n=-1,a_{n}=\frac12$

Not a polynomial function.

Verified step by step guidance

Step 1: Recall the definition of a polynomial function. A polynomial function is a sum of terms, each of which is a constant multiplied by a variable raised to a non-negative integer power. For example, terms like 4x^3 and -2x are valid, but terms like x^(-1) or fractional exponents are not allowed in a polynomial.

Step 2: Analyze the given function f(x) = 4x^3 + (1/2)x^(-1) - 2x + 1. Identify each term and its corresponding exponent. The terms are: 4x^3 (exponent = 3), (1/2)x^(-1) (exponent = -1), -2x (exponent = 1), and 1 (constant term, exponent = 0).

Step 3: Check if all exponents in the function are non-negative integers. In this case, the term (1/2)x^(-1) has an exponent of -1, which is not a non-negative integer. This violates the definition of a polynomial function.

Step 4: Conclude that the given function is not a polynomial function because it contains a term with a negative exponent, specifically (1/2)x^(-1).

Step 5: If the function were a polynomial, we would write it in standard form (arranging terms in descending order of exponents), identify the degree (highest exponent), and the leading coefficient (coefficient of the term with the highest exponent). However, since this is not a polynomial, these steps are not applicable.