To determine whether each of the given tables represents a linear function, we must examine whether the difference between the corresponding [tex]\( y \)[/tex]-values ([tex]\( g(x) \)[/tex], [tex]\( f(x) \)[/tex], [tex]\( h(x) \)[/tex], or [tex]\( k(x) \)[/tex]) is consistent for equal differences in [tex]\( x \)[/tex]-values. We base our conclusion on this consistency.
Let's evaluate each table separately:
1. Table for [tex]\( g(x) \)[/tex]:
[tex]\[
\begin{array}{|r|r|}
\hline
x & g(x) \\
\hline
0 & 5 \\
\hline
5 & -10 \\
\hline
10 & -25 \\
\hline
15 & -40 \\
\hline
\end{array}
\][/tex]
- The difference for [tex]\( x \)[/tex] values of 0 to 5 is [tex]\( -10 - 5 = -15 \)[/tex].
- The difference for [tex]\( x \)[/tex] values of 5 to 10 is [tex]\( -25 - (-10) = -15 \)[/tex].
- The difference for [tex]\( x \)[/tex] values of 10 to 15 is [tex]\( -40 - (-25) = -15 \)[/tex].
Since the differences are consistent, [tex]\( g(x) \)[/tex] could represent a linear function.
2. Table for [tex]\( f(x) \)[/tex]:
[tex]\[
\begin{array}{|r|r|}
\hline
x & f(x) \\
\hline
0 & -4 \\
\hline
5 & 11 \\
\hline
10 & 26 \\
\hline
15 & 41 \\
\hline
\end{array}
\][/tex]
- The difference for [tex]\( x \)[/tex] values of 0 to 5 is [tex]\( 11 - (-4) = 15 \)[/tex].
- The difference for [tex]\( x \)[/tex] values of 5 to 10 is [tex]\( 26 - 11 = 15 \)[/tex].
- The difference for [tex]\( x \)[/tex] values of 10 to 15 is [tex]\( 41 - 26 = 15 \)[/tex].
Since the differences are consistent, [tex]\( f(x) \)[/tex] could represent a linear function.
3. Table for [tex]\( h(x) \)[/tex]:
[tex]\[
\begin{array}{|r|r|}
\hline
x & h(x) \\
\hline
0 & 5 \\
\hline
5 & 30 \\
\hline
10 & 105 \\
\hline
15 & 230 \\
\hline
\end{array}
\][/tex]
- The difference for [tex]\( x \)[/tex] values of 0 to 5 is [tex]\( 30 - 5 = 25 \)[/tex].
- The difference for [tex]\( x \)[/tex] values of 5 to 10 is [tex]\( 105 - 30 = 75 \)[/tex].
- The difference for [tex]\( x \)[/tex] values of 10 to 15 is [tex]\( 230 - 105 = 125 \)[/tex].
Since the differences are not consistent, [tex]\( h(x) \)[/tex] cannot represent a linear function.
4. Table for [tex]\( k(x) \)[/tex]:
[tex]\[
\begin{array}{|r|r|}
\hline
x & k(x) \\
\hline
5 & 11 \\
\hline
10 & 26 \\
\hline
20 & 56 \\
\hline
25 & 71 \\
\hline
\end{array}
\][/tex]
- The difference for [tex]\( x \)[/tex] values of 5 to 10 is [tex]\( 26 - 11 = 15 \)[/tex].
- The difference for [tex]\( x \)[/tex] values of 10 to 20 is [tex]\( 56 - 26 = 30 \)[/tex].
- The difference for [tex]\( x \)[/tex] values of 20 to 25 is [tex]\( 71 - 56 = 15 \)[/tex].
Here, the differences are not consistent, indicating that [tex]\( k(x) \)[/tex] cannot represent a linear function.
Based on our analysis:
- The table for [tex]\( g(x) \)[/tex] represents a linear function.
- The table for [tex]\( f(x) \)[/tex] represents a linear function.
- The table for [tex]\( h(x) \)[/tex] does not represent a linear function.
- The table for [tex]\( k(x) \)[/tex] does not represent a linear function.
Therefore, the tables that could represent a linear function are the ones for [tex]\( g(x) \)[/tex] and [tex]\( f(x) \)[/tex].
