Analyze the relationship in this table, and look for a pattern that relates [tex] x [/tex] and [tex] f(x) [/tex].

\begin{tabular}{|c|c|}
\hline
[tex] x [/tex] & [tex] f(x) [/tex] \\
\hline
1 & 4 \\
4 & 8 \\
9 & 12 \\
16 & 16 \\
25 & 20 \\
\hline
\end{tabular}

Part A:
Identify the function type that could represent the data in the table. Explain your reasoning.

Type your response in the box.



Answer :

To analyze the relationship between [tex]\( x \)[/tex] and [tex]\( f(x) \)[/tex] based on the given data, we can calculate the difference between each pair of [tex]\( x \)[/tex] and [tex]\( f(x) \)[/tex] values.

First, let's compute the differences:
- For [tex]\( x = 1 \)[/tex] and [tex]\( f(x) = 4 \)[/tex]:
[tex]\[ f(x) - x = 4 - 1 = 3 \][/tex]

- For [tex]\( x = 4 \)[/tex] and [tex]\( f(x) = 8 \)[/tex]:
[tex]\[ f(x) - x = 8 - 4 = 4 \][/tex]

- For [tex]\( x = 9 \)[/tex] and [tex]\( f(x) = 12 \)[/tex]:
[tex]\[ f(x) - x = 12 - 9 = 3 \][/tex]

- For [tex]\( x = 16 \)[/tex] and [tex]\( f(x) = 16 \)[/tex]:
[tex]\[ f(x) - x = 16 - 16 = 0 \][/tex]

- For [tex]\( x = 25 \)[/tex] and [tex]\( f(x) = 20 \)[/tex]:
[tex]\[ f(x) - x = 20 - 25 = -5 \][/tex]

Collecting all the differences, we have:
[tex]\[ [3, 4, 3, 0, -5] \][/tex]

Next, we check if these differences are consistent. If the differences were consistent, it would imply a linear relationship where [tex]\( f(x) \)[/tex] changes by a constant amount relative to [tex]\( x \)[/tex]. However, in this case, the differences are not consistent. We have:
[tex]\[ 3, 4, 3, 0, -5 \][/tex]

Because the differences vary and are not constant, it suggests that the relationship between [tex]\( x \)[/tex] and [tex]\( f(x) \)[/tex] is not linear.

Therefore, the type of function that could represent this data is likely non-linear. The inconsistencies in the differences indicate that the relationship does not align with a simple linear function.