To determine which function rule correctly models the function over the given domain, we'll test each option with the provided [tex]\( x \)[/tex] values and their corresponding [tex]\( f(x) \)[/tex] values.
The given data is:
[tex]\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
-7 & -11 \\
\hline
-1 & 1 \\
\hline
3 & 9 \\
\hline
4 & 11 \\
\hline
7 & 17 \\
\hline
\end{array}
\][/tex]
We need to check which function rule among the given options satisfies these pairs of [tex]\((x, f(x))\)[/tex]. The function rules are:
1. [tex]\( f(x) = 3x + 10 \)[/tex]
2. [tex]\( f(x) = 2x + 3 \)[/tex]
3. [tex]\( f(x) = 4x + 5 \)[/tex]
4. [tex]\( f(x) = 3x - 10 \)[/tex]
We will substitute each [tex]\( x \)[/tex] value into each function and see if it matches the corresponding [tex]\( f(x) \)[/tex] value.
### Option 1: [tex]\( f(x) = 3x + 10 \)[/tex]
- For [tex]\( x = -7 \)[/tex]: [tex]\( f(-7) = 3(-7) + 10 = -21 + 10 = -11 \)[/tex] (matches)
- For [tex]\( x = -1 \)[/tex]: [tex]\( f(-1) = 3(-1) + 10 = -3 + 10 = 7 \)[/tex] (does not match)
- The mismatch with the value at [tex]\( x=-1 \)[/tex] means this rule is not correct.
### Option 2: [tex]\( f(x) = 2x + 3 \)[/tex]
- For [tex]\( x = -7 \)[/tex]: [tex]\( f(-7) = 2(-7) + 3 = -14 + 3 = -11 \)[/tex] (matches)
- For [tex]\( x = -1 \)[/tex]: [tex]\( f(-1) = 2(-1) + 3 = -2 + 3 = 1 \)[/tex] (matches)
- For [tex]\( x = 3 \)[/tex]: [tex]\( f(3) = 2(3) + 3 = 6 + 3 = 9 \)[/tex] (matches)
- For [tex]\( x = 4 \)[/tex]: [tex]\( f(4) = 2(4) + 3 = 8 + 3 = 11 \)[/tex] (matches)
- For [tex]\( x = 7 \)[/tex]: [tex]\( f(7) = 2(7) + 3 = 14 + 3 = 17 \)[/tex] (matches)
Since all pairs [tex]\((x, f(x))\)[/tex] match with the rule [tex]\( f(x) = 2x + 3 \)[/tex], this rule is correct.
### Option 3: [tex]\( f(x) = 4x + 5 \)[/tex]
- For [tex]\( x = -7 \)[/tex]: [tex]\( f(-7) = 4(-7) + 5 = -28 + 5 = -23 \)[/tex] (does not match)
- The mismatch means this rule is not correct.
### Option 4: [tex]\( f(x) = 3x - 10 \)[/tex]
- For [tex]\( x = -7 \)[/tex]: [tex]\( f(-7) = 3(-7) - 10 = -21 - 10 = -31 \)[/tex] (does not match)
- The mismatch means this rule is not correct.
After examining each function, we find that the function rule that accurately models the given function over the specified domain is:
[tex]\[ f(x) = 2x + 3 \][/tex]
Therefore, the correct function rule is modeled by option 2.
