To determine the type of function that best describes [tex]\( g(x) \)[/tex] given the table of values, we proceed as follows:
[tex]\[
\begin{array}{|c|c|c|}
\hline
x & g(x) \\
\hline
-3 & 4 \\
-1 & -4 \\
1 & -4 \\
3 & 4 \\
5 & 20 \\
7 & 44 \\
9 & 76 \\
\hline
\end{array}
\][/tex]
We are asked to determine whether the function [tex]\( g(x) \)[/tex] is Exponential, Logarithmic, Polynomial, or Rational. Let's analyze each possibility:
1. Exponential Function:
- An exponential function generally has the form [tex]\( f(x) = a \cdot b^x \)[/tex], where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are constants.
- Exponential functions typically exhibit rapid growth (or decay). The values of [tex]\( g(x) \)[/tex] change sign and do not exhibit the consistent multiplicative growth characteristic of exponential functions.
2. Logarithmic Function:
- A logarithmic function has the form [tex]\( f(x) = a \cdot \log_b(x) + c \)[/tex], where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are constants.
- Logarithmic functions usually grow more slowly and have a characteristic curve which is not reflected in the given values of [tex]\( g(x) \)[/tex]. Moreover, logarithmic functions are undefined for negative [tex]\( x \)[/tex]-values without considering transformations, which is not suitable given the mixed signs and values in the dataset.
3. Polynomial Function:
- A polynomial function has the form [tex]\( f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \)[/tex].
- Polynomial functions can describe various patterns of data changes, including changing directions multiple times (as seen with varying signs and values of [tex]\( g(x) \)[/tex]). The given values appearing for [tex]\( g(x) \)[/tex] exhibit behavior that might fit a polynomial of a particular degree.
4. Rational Function:
- A rational function is the ratio of two polynomials, [tex]\( f(x) = \frac{p(x)}{q(x)} \)[/tex].
- While rational functions can describe a variety of trends, including discontinuities and complex behaviors, the specific values given in [tex]\( g(x) \)[/tex] don’t immediately suggest a typical rational function’s behavior given more conventional fits for polynomial functions.
Through these analyses and matching characteristics:
From the given table and after examining the values, it can be inferred that the function that best fits the provided data pattern is a polynomial function.
Thus, the type of function that describes [tex]\( g(x) \)[/tex] is:
[tex]\[
\boxed{\text{Polynomial}}
\][/tex]
