Question

Determine if the following function is a polynomial function: f() = ²-3 / ⁴

Answer 1

The given function
`\(f(x) = (x^2 - 3)/(x^4)\)` is not a polynomial function.

Step-by-step explanation:

A polynomial function is a mathematical function of the form
`\(f(x) = a_nx^n + a_(n-1)x^(n-1) + \ldots + a_1x + a_0\)`, where
`\(a_n, a_(n-1), \ldots, a_1, a_0\)` are constants, and n is a non-negative integer. In this case, the given function involves a fraction with a variable in the denominator raised to a power of 4, making it a rational function rather than a polynomial.

The function can be expressed as
`\(f(x) = (x^2 - 3)/(x^4) = (1)/(x^2) - (3)/(x^4)\)`, where the terms involve negative exponents. A polynomial function only allows non-negative integer exponents on the variable. Therefore, the given function is not a polynomial function.

While the numerator is a polynomial of degree 2, the presence of
`\(x^(-2)\)` and
`\(x^(-4)\)` terms in the denominator makes the function a rational function. Understanding the form of polynomial functions and recognizing the constraints on exponents helps classify mathematical functions accurately.