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.
