To determine which function is most likely growing exponentially, let’s examine the patterns of growth in the provided values for [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] at various points [tex]\( x \)[/tex].
Looking at the table:
[tex]\[
\begin{array}{|c|c|c|}
\hline
x & f(x) & g(x) \\
\hline
1 & 2 & 2 \\
\hline
2 & 5 & 4 \\
\hline
3 & 10 & 8 \\
\hline
4 & 17 & 16 \\
\hline
5 & 26 & 32 \\
\hline
\end{array}
\][/tex]
First, let's identify the rate at which [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are growing:
### For [tex]\( f(x) \)[/tex]:
- At [tex]\( x = 1 \)[/tex], [tex]\( f(x) = 2 \)[/tex]
- At [tex]\( x = 2 \)[/tex], [tex]\( f(x) = 5 \)[/tex]
- At [tex]\( x = 3 \)[/tex], [tex]\( f(x) = 10 \)[/tex]
- At [tex]\( x = 4 \)[/tex], [tex]\( f(x) = 17 \)[/tex]
- At [tex]\( x = 5 \)[/tex], [tex]\( f(x) = 26 \)[/tex]
The differences in consecutive [tex]\( f(x) \)[/tex] values are:
[tex]\[
f(2) - f(1) = 5 - 2 = 3 \\
f(3) - f(2) = 10 - 5 = 5 \\
f(4) - f(3) = 17 - 10 = 7 \\
f(5) - f(4) = 26 - 17 = 9 \\
\][/tex]
The differences (3, 5, 7, 9) suggest a quadratic pattern, as they increase linearly.
### For [tex]\( g(x) \)[/tex]:
- At [tex]\( x = 1 \)[/tex], [tex]\( g(x) = 2 \)[/tex]
- At [tex]\( x = 2 \)[/tex], [tex]\( g(x) = 4 \)[/tex]
- At [tex]\( x = 3 \)[/tex], [tex]\( g(x) = 8 \)[/tex]
- At [tex]\( x = 4 \)[/tex], [tex]\( g(x) = 16 \)[/tex]
- At [tex]\( x = 5 \)[/tex], [tex]\( g(x) = 32 \)[/tex]
The differences in consecutive [tex]\( g(x) \)[/tex] values are:
[tex]\[
g(2) - g(1) = 4 - 2 = 2 \\
g(3) - g(2) = 8 - 4 = 4 \\
g(4) - g(3) = 16 - 8 = 8 \\
g(5) - g(4) = 32 - 16 = 16 \\
\][/tex]
The differences (2, 4, 8, 16) suggest an exponential pattern since each difference is a multiple of the previous.
### Conclusion:
From these observations, [tex]\( g(x) \)[/tex] is the function that grows faster and is most likely the exponential function due to its rapid and multiplicative increase.
Thus, the most likely answer is:
[tex]\[
g(x), \text{ because it grows faster than } f(x).
\][/tex]
