To determine which of the three functions [tex]\( f(x) \)[/tex], [tex]\( g(x) \)[/tex], and [tex]\( h(x) \)[/tex] represent linear relationships, we need to analyze their values and check for a constant rate of change (slope).
### Function [tex]\( f(x) \)[/tex]
[tex]\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
1 & 5 \\
\hline
2 & 11 \\
\hline
3 & 29 \\
\hline
4 & 83 \\
\hline
\end{array}
\][/tex]
To check for linearity, we look at the differences between consecutive [tex]\( x \)[/tex] and [tex]\( f(x) \)[/tex] values:
For [tex]\( x \)[/tex]:
[tex]\[
\Delta x = 1 - 1 = 1, \quad 2 - 1 = 1, \quad 3 - 2 = 1, \quad 4 - 3 = 1
\][/tex]
For [tex]\( f(x) \)[/tex]:
[tex]\[
\Delta f(x) = 11 - 5 = 6, \quad 29 - 11 = 18, \quad 83 - 29 = 54
\][/tex]
The differences are not consistent, indicating that [tex]\( f(x) \)[/tex] is not linear.
### Function [tex]\( g(x) \)[/tex]
[tex]\[
\begin{array}{|c|c|}
\hline
x & g(x) \\
\hline
3 & 17 \\
\hline
5 & 25 \\
\hline
7 & 33 \\
\hline
9 & 41 \\
\hline
\end{array}
\][/tex]
For [tex]\( x \)[/tex]:
[tex]\[
\Delta x = 5 - 3 = 2, \quad 7 - 5 = 2, \quad 9 - 7 = 2
\][/tex]
For [tex]\( g(x) \)[/tex]:
[tex]\[
\Delta g(x) = 25 - 17 = 8, \quad 33 - 25 = 8, \quad 41 - 33 = 8
\][/tex]
The differences are consistent, indicating that [tex]\( g(x) \)[/tex] is linear.
### Function [tex]\( h(x) \)[/tex]
[tex]\[
\begin{array}{|c|c|}
\hline
x & h(x) \\
\hline
2 & 8.8 \\
\hline
4 & 15.6 \\
\hline
6 & 22.4 \\
\hline
8 & 29.2 \\
\hline
\end{array}
\][/tex]
For [tex]\( x \)[/tex]:
[tex]\[
\Delta x = 4 - 2 = 2, \quad 6 - 4 = 2, \quad 8 - 6 = 2
\][/tex]
For [tex]\( h(x) \)[/tex]:
[tex]\[
\Delta h(x) = 15.6 - 8.8 = 6.8, \quad 22.4 - 15.6 = 6.8, \quad 29.2 - 22.4 = 6.8
\][/tex]
The differences are not integer multiples of a common value;
Therefore, the only function that represents a linear relationship is:
[tex]\(\boxed{g(x)}\)[/tex]
