Answer :
To determine which function is not a polynomial, let's review the characteristics of a polynomial function:
A polynomial function is an expression that can be written in the form:
[tex]\[ f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \][/tex]
where [tex]\( a_n, a_{n-1}, \ldots, a_1, a_0 \)[/tex] are constants and [tex]\( n \)[/tex] is a non-negative integer. The exponents of the variable [tex]\( x \)[/tex] in a polynomial must be whole numbers (0, 1, 2, etc.).
Now let's evaluate each of the given functions:
1. [tex]\( f(x) = 4|x+2| - 5 \)[/tex]
- This function includes an absolute value term, [tex]\( |x+2| \)[/tex]. The presence of an absolute value means that the function is not a polynomial. Polynomial functions do not have absolute values because they need to be written as sums of powers of [tex]\( x \)[/tex] with constant coefficients.
2. [tex]\( f(x) = -\frac{1}{4} x^3 + x^2 - 4 \)[/tex]
- This is a polynomial function because it can be written with terms [tex]\( -\frac{1}{4} x^3 \)[/tex], [tex]\( x^2 \)[/tex], and a constant [tex]\(-4\)[/tex]. All exponents are non-negative integers, making it a polynomial.
3. [tex]\( f(x) = -\frac{1}{2} x + 3 \)[/tex]
- This is a polynomial function. It is a linear polynomial because it can be written in the form [tex]\( ax + b \)[/tex], with the exponent of [tex]\( x \)[/tex] being 1.
4. [tex]\( f(x) = 2x + 7 + 3x - 1 \)[/tex]
- After combining like terms, this simplifies to [tex]\( 5x + 6 \)[/tex], which is a linear polynomial. The exponents remain whole numbers, making it a polynomial function.
Therefore, among the given options, the function [tex]\( f(x) = 4|x+2| - 5 \)[/tex] is the one that is not a polynomial function.
A polynomial function is an expression that can be written in the form:
[tex]\[ f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \][/tex]
where [tex]\( a_n, a_{n-1}, \ldots, a_1, a_0 \)[/tex] are constants and [tex]\( n \)[/tex] is a non-negative integer. The exponents of the variable [tex]\( x \)[/tex] in a polynomial must be whole numbers (0, 1, 2, etc.).
Now let's evaluate each of the given functions:
1. [tex]\( f(x) = 4|x+2| - 5 \)[/tex]
- This function includes an absolute value term, [tex]\( |x+2| \)[/tex]. The presence of an absolute value means that the function is not a polynomial. Polynomial functions do not have absolute values because they need to be written as sums of powers of [tex]\( x \)[/tex] with constant coefficients.
2. [tex]\( f(x) = -\frac{1}{4} x^3 + x^2 - 4 \)[/tex]
- This is a polynomial function because it can be written with terms [tex]\( -\frac{1}{4} x^3 \)[/tex], [tex]\( x^2 \)[/tex], and a constant [tex]\(-4\)[/tex]. All exponents are non-negative integers, making it a polynomial.
3. [tex]\( f(x) = -\frac{1}{2} x + 3 \)[/tex]
- This is a polynomial function. It is a linear polynomial because it can be written in the form [tex]\( ax + b \)[/tex], with the exponent of [tex]\( x \)[/tex] being 1.
4. [tex]\( f(x) = 2x + 7 + 3x - 1 \)[/tex]
- After combining like terms, this simplifies to [tex]\( 5x + 6 \)[/tex], which is a linear polynomial. The exponents remain whole numbers, making it a polynomial function.
Therefore, among the given options, the function [tex]\( f(x) = 4|x+2| - 5 \)[/tex] is the one that is not a polynomial function.