To determine the linear function [tex]\( f(x) \)[/tex] that fits the given table of values, we need to find a function of the form [tex]\( f(x) = ax + b \)[/tex], where [tex]\( a \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
We are given the following values:
[tex]\[
\begin{array}{c|cccc}
x & 0 & 1 & 2 & 3 \\
\hline
f(x) & -1 & 2 & 5 & 8 \\
\end{array}
\][/tex]
### Step-by-Step Solution
1. Identify the x and y values:
[tex]\[
x: \{0, 1, 2, 3\}
\][/tex]
[tex]\[
f(x): \{-1, 2, 5, 8\}
\][/tex]
2. Establish the system of equations using the linear form [tex]\( f(x) = ax + b \)[/tex]:
For [tex]\( x = 0 \)[/tex], [tex]\( f(0) = -1 \)[/tex]:
[tex]\[
a \cdot 0 + b = -1 \quad \Rightarrow \quad b = -1
\][/tex]
For [tex]\( x = 1 \)[/tex], [tex]\( f(1) = 2 \)[/tex]:
[tex]\[
a \cdot 1 + b = 2 \quad \Rightarrow \quad a + b = 2 \quad \Rightarrow \quad a + (-1) = 2 \quad \Rightarrow \quad a = 3
\][/tex]
3. Using these coefficients [tex]\( a \)[/tex] and [tex]\( b \)[/tex], express the linear function:
[tex]\[
f(x) = 3x - 1
\][/tex]
4. Verify with the remaining points [tex]\( (2, 5) \)[/tex] and [tex]\( (3, 8) \)[/tex] to ensure the function fits all given data, although we can be confident as we've obtained general coefficients useful for the function:
For [tex]\( x = 2 \)[/tex], [tex]\( f(2) = 3 \cdot 2 - 1 = 6 - 1 = 5 \)[/tex]
For [tex]\( x = 3 \)[/tex], [tex]\( f(3) = 3 \cdot 3 - 1 = 9 - 1 = 8 \)[/tex]
Both of these results match the table values, confirming that our function is correct.
Therefore, the linear function [tex]\( f(x) \)[/tex] that fits the given table of values is:
[tex]\[
f(x) = 3x - 1
\][/tex]
