Let's examine the given functions using their tabulated values.
For [tex]\( p(x) \)[/tex], the table is:
[tex]\[
\begin{tabular}{|c|c|}
\hline$x$ & $p(x)$ \\
\hline 0 & 4 \\
\hline 1 & 16 \\
\hline 2 & 64 \\
\hline
\end{tabular}
\][/tex]
We can observe that the values of [tex]\( p(x) \)[/tex] follow a specific pattern:
- When [tex]\( x = 0 \)[/tex], [tex]\( p(x) = 4 \)[/tex].
- When [tex]\( x = 1 \)[/tex], [tex]\( p(x) = 16 \)[/tex].
- When [tex]\( x = 2 \)[/tex], [tex]\( p(x) = 64 \)[/tex].
If we observe carefully, each value of [tex]\( p(x) \)[/tex] seems to multiply by 4 as [tex]\( x \)[/tex] increases by 1:
[tex]\[ 4 \cdot 4 = 16 \][/tex]
[tex]\[ 16 \cdot 4 = 64 \][/tex]
This indicates that [tex]\( p(x) \)[/tex] is an exponential function of the form:
[tex]\[ p(x) = 4 \cdot 4^x \][/tex]
Thus, [tex]\( p(x) \)[/tex] is both nonlinear and exponential.
Next, let’s analyze [tex]\( q(x) \)[/tex] from its table:
[tex]\[
\begin{tabular}{|c|c|}
\hline$x$ & $q(x)$ \\
\hline 0 & 3 \\
\hline 2 & 15 \\
\hline 5 & 33 \\
\hline
\end{tabular}
\][/tex]
The increments in [tex]\( x \)[/tex] are not uniform which complicates the analysis. Let's look at the change in [tex]\( q(x) \)[/tex]:
- From [tex]\( x = 0 \)[/tex] to [tex]\( x = 2 \)[/tex], [tex]\( q(x) \)[/tex] changes from 3 to 15.
- From [tex]\( x = 2 \)[/tex] to [tex]\( x = 5 \)[/tex], [tex]\( q(x) \)[/tex] changes from 15 to 33.
The changes are not consistent with a linear pattern since the increments are different:
[tex]\[ q(2) - q(0) = 15 - 3 = 12 \][/tex]
[tex]\[ q(5) - q(2) = 33 - 15 = 18 \][/tex]
Also, there's no evident multiplicative pattern indicating it's not exponential. Therefore, [tex]\( q(x) \)[/tex] is neither linear nor exponential, meaning it is a general nonlinear function.
Based on these observations:
- [tex]\( p(x) \)[/tex] is nonlinear and follows an exponential pattern.
- [tex]\( q(x) \)[/tex] does not follow a linear or exponential pattern.
Thus, the correct statement describing these functions is:
A. Both functions are nonlinear.
This matches our detailed analysis of the functions [tex]\( p(x) \)[/tex] and [tex]\( q(x) \)[/tex].
