Question

Identify the function as a power function, a polynomial function, or neither

(a) f(x) = x - x⁴
(b) f(x) = 2x (x + 2)(x - 1)²
(c) f(x) = (x²)³
(d) f(x) = x²/ x²-1
(e) f(x) = 3x+1

Answer 1

Function (a) is a polynomial function, (b) is also a polynomial function, (c) is a power function, (d) is neither (a rational function), and (e) is a polynomial function.

Step-by-step explanation:

Let's identify each function given in the question:

• Power functions are of the form f(x) = x^n where n is a real number.
• Polynomial functions are sums of terms of the form ax^n where n is a non-negative integer, and a is a real number.
• Functions that do not fit the above forms are neither.

1. f(x) = x - x⁴ is a polynomial function because it is a sum of terms x^1 and x^4.
2. f(x) = 2x (x + 2)(x - 1)² is also a polynomial function because when expanded, it is a sum of terms each in the form of ax^n.
3. f(x) = (x²)³ is a power function because it simplifies to x^6, which is of the form x^n.
4. f(x) = x² / (x²-1) is neither because the presence of x² in the denominator makes it a rational function, not a polynomial or power function.
5. f(x) = 3x+1 is a polynomial function because it is a linear polynomial of degree one, which fits the form of polynomial functions.