Problem 20 - Precalculus : Building Concepts and Connections

Question

Determine whether the function is a polynomial function. If so, find the degree. If not, state the reason. $$g(x)=-2$$

Step-by-Step Solution

Answer

Yes, the function \(g(x) = -2\) is a polynomial function, and its degree is 0.

Step 1: Identify the Function

The function that we are analyzing is \(g(x) = -2\). This is a constant function, which means that the output value doesn't change no matter what x-value is put into it.

Step 2: Determine If It's a Polynomial

A polynomial is an equation of the form \[ ax^n + bx^{n-1} + ... + k = 0 \] where a, b, ... , k are constants, n is a non-negative integer and x is the variable. In our case, since there are no variables in the function, it can still be considered a polynomial function as it can be written in the format of \(ax^n\), where \(a = -2\), \(n = 0\), and \(x\) is the variable.

Step 3: Identify the Degree of the Polynomial

The degree of a polynomial is the greatest exponent in the polynomial. Since in our function, the implied exponent of x is 0 (as there is no x variable present, we can think of it as \(x^0\), which equals to 1), the degree of our polynomial is 0.

Key Concepts

Constant Function

A constant function is a type of function where the value remains the same regardless of the input. Essentially, no matter what variable value you plug into the equation, the output will always be the same. In mathematical terms, a constant function has the form
\( f(x) = c \)
where \( c \) is a constant. Constant functions are considered the simplest kinds of polynomial functions, with their graphs being horizontal lines on the coordinate plane. For instance, the function
\( g(x) = -2 \)
represents a constant function because it denotes the same output, -2, no matter the value of x. This simplicity, however, doesn't take away from the importance of understanding constant functions as they form the basis for more complex polynomial equations.

Polynomial Equation

A polynomial equation is typically expressed in the form
\[ ax^n + bx^{n-1} + \ldots + k = 0 \]
where the letters \( a, b, \ldots, k \) represent constants, \( n \) is a non-negative integer reflecting the highest degree of the variable \( x \), and each term is a power of the variable x. What categorizes an equation as polynomial is the presence of whole number exponents (including zero) and the addition and subtraction of these terms. Variables in a polynomial equation can only be multiplied by constants or raised to positive integer powers—they cannot appear in the denominator, nor can they have negative or fractional exponents. In the example of the function
\( g(x) = -2 \),
while there is no x-variable present, it fits into the broader category of polynomial equations because it represents a polynomial of degree zero.

Degree of a Polynomial

The degree of a polynomial is determined by the highest power of the variable x in the equation. It provides a way to classify polynomials and to predict the behavior of their graphs. For example, quadratic polynomials have a degree of 2 and typically show a parabolic shape when graphed. In mathematical terms, the degree is found by looking at
\( ax^n \)
which is the term with the greatest exponent \( n \). A higher degree implies a more complex graph with more turns and intersections with the x-axis. In our case, such as with the function
\( g(x) = -2 \),
the degree is 0 because we interpret the polynomial as \( -2 \times x^0 \), and as any number to the power of zero is 1, it does not affect the value of the constant -2. The degree of zero indicates that the graph is a flat line, which is characteristic of constant functions.