To determine which of the given tables represents an exponential function, let's recall that an exponential function can be defined as [tex]\( f(x) = a \cdot b^x \)[/tex], where [tex]\( a \)[/tex] is a constant and [tex]\( b \)[/tex] is the base of the exponential function.
We should check if the [tex]\( f(x) \)[/tex] values grow by a consistent multiplicative factor. Let's examine each table in detail:
### Table 1:
[tex]\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
0 & 1 \\
\hline
1 & 3 \\
\hline
2 & 5 \\
\hline
3 & 8 \\
\hline
4 & 11 \\
\hline
\end{array}
\][/tex]
- [tex]\( f(1) / f(0) = 3 / 1 = 3 \)[/tex]
- [tex]\( f(2) / f(1) = 5 / 3 \approx 1.67 \)[/tex]
- [tex]\( f(3) / f(2) = 8 / 5 = 1.6 \)[/tex]
- [tex]\( f(4) / f(3) = 11 / 8 \approx 1.375 \)[/tex]
The growth factor is not consistent. Hence, Table 1 does not represent an exponential function.
### Table 2:
[tex]\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
0 & 1 \\
\hline
1 & 4 \\
\hline
2 & 16 \\
\hline
3 & 64 \\
\hline
4 & 256 \\
\hline
\end{array}
\][/tex]
- [tex]\( f(1) = 4 = 4^1 \)[/tex]
- [tex]\( f(2) = 16 = 4^2 \)[/tex]
- [tex]\( f(3) = 64 = 4^3 \)[/tex]
- [tex]\( f(4) = 256 = 4^4 \)[/tex]
The pattern shows that [tex]\( f(x) = 4^x \)[/tex]. This indicates that the values grow by a base of 4 for each increment in [tex]\( x \)[/tex], which is a consistent exponential growth factor.
### Table 3:
[tex]\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
0 & 2 \\
\hline
1 & 4 \\
\hline
2 & 6 \\
\hline
3 & 10 \\
\hline
4 & 12 \\
\hline
\end{array}
\][/tex]
- [tex]\( f(1) / f(0) = 4 / 2 = 2 \)[/tex]
- [tex]\( f(2) / f(1) = 6 / 4 = 1.5 \)[/tex]
- [tex]\( f(3) / f(2) = 10 / 6 \approx 1.67 \)[/tex]
- [tex]\( f(4) / f(3) = 12 / 10 = 1.2 \)[/tex]
The growth factor again is not consistent. Thus, Table 3 does not represent an exponential function.
Conclusion: Based on the consistent exponential pattern we see in the values of [tex]\( f(x) \)[/tex] in Table 2, it is evident that Table 2 represents an exponential function.
