Let's analyze the given data points and determine the nature of the function, along with its explicit and recursive equations.
We have the following data points:
[tex]\[
\begin{array}{c|c}
x & f(x) \\
\hline
6 & 11 \\
7 & 13 \\
8 & 15 \\
9 & 17 \\
10 & 19 \\
\end{array}
\][/tex]
### Step-by-Step Solution:
1. Determine the differences between consecutive [tex]\( f(x) \)[/tex] values:
Calculate the differences between each consecutive pair of [tex]\( f(x) \)[/tex] values to see if the change is constant.
[tex]\[
\begin{aligned}
f(7) - f(6) &= 13 - 11 = 2 \\
f(8) - f(7) &= 15 - 13 = 2 \\
f(9) - f(8) &= 17 - 15 = 2 \\
f(10) - f(9) &= 19 - 17 = 2 \\
\end{aligned}
\][/tex]
Since all differences are equal to 2, the function [tex]\( f(x) \)[/tex] shows a constant rate of change.
2. Identify the type of function:
A constant rate of change in the differences indicates that [tex]\( f(x) \)[/tex] is a linear function.
3. Find the explicit equation of the linear function:
The general form of a linear function is:
[tex]\[
f(x) = mx + b
\][/tex]
where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
From the differences, we know that the slope [tex]\( m \)[/tex] is 2.
Now, use one of the points to solve for [tex]\( b \)[/tex]. Using [tex]\( x = 6 \)[/tex] and [tex]\( f(6) = 11 \)[/tex]:
[tex]\[
11 = 2 \cdot 6 + b \\
11 = 12 + b \\
b = 11 - 12 \\
b = -1
\][/tex]
Therefore, the explicit equation is:
[tex]\[
f(x) = 2x - 1
\][/tex]
4. Find the recursive equation:
The recursive form of a linear function can be expressed as:
[tex]\[
f(x + 1) = f(x) + m
\][/tex]
Given [tex]\( m = 2 \)[/tex], the recursive equation is:
[tex]\[
f(x + 1) = f(x) + 2
\][/tex]
### Summary:
- Type of function and nature of change:
The function is linear with a constant rate of change of 2.
- Explicit equation:
[tex]\[
f(x) = 2x - 1
\][/tex]
- Recursive equation:
[tex]\[
f(x + 1) = f(x) + 2
\][/tex]
