To determine the type of function that describes [tex]\( g(x) \)[/tex], we can analyze the given table of values for [tex]\( x \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline
x & -3 & -1 & 1 & 3 & 5 & 7 & 9 \\
\hline
g(x) & 4 & -4 & -4 & 4 & 20 & 44 & 76 \\
\hline
\end{array}
\][/tex]
First, let's consider each of the given options and see which best fits the pattern in the values:
1. Exponential Function: An exponential function has the general form [tex]\( g(x) = a \cdot b^x \)[/tex]. In an exponential function, the values of [tex]\( g(x) \)[/tex] would grow or decay at a consistent multiplicative rate, which doesn't seem to fit the values provided here.
2. Logarithmic Function: A logarithmic function typically has the general form [tex]\( g(x) = a \cdot \log_b(x) + c \)[/tex]. These functions increase or decrease slowly compared to polynomial or exponential types. The values provided do not suggest the slow, consistent growth or decline typical of logarithmic functions.
3. Polynomial Function: A polynomial function is of the form [tex]\( g(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 \)[/tex]. This type of function can fit a wide variety of patterns depending on the degree of the polynomial (the highest power of [tex]\( x \)[/tex]). Given the variety and specific variation in [tex]\( g(x) \)[/tex], such a function seems plausible.
4. Rational Functions: A rational function is a ratio of two polynomials, [tex]\( g(x) = \frac{p(x)}{q(x)} \)[/tex]. For the values to fit this type, we would expect potential asymptotic behavior or values that aren't as smoothly fitting as polynomials or other types. The behavior in the table doesn't suggest a rational function's complexity.
Given the complexity and direction changes in the [tex]\( g(x) \)[/tex] values, let's analyze the polynomial option more closely.
Upon fitting the given values to a polynomial function, the coefficients that fit the values nearly perfectly were found to be consistent with a polynomial. The specific coefficients were:
[tex]\[ g(x) = -6.37719795 \times 10^{-18} x^3 + 1 x^2 - 2.14847929 \times 10^{-15} x - 5 \][/tex]
The polynomial fit values were extremely close to the original values, with the maximum difference being extremely small ([tex]\( \approx 2.842170943040401 \times 10^{-14} \)[/tex]), which confirms a polynomial function closely matches the [tex]\( g(x) \)[/tex] values given.
Thus, the type of function that describes [tex]\( g(x) \)[/tex] based on the table values is best described as:
Polynomial
