To determine which function was used to generate the given values of [tex]\( f(x) \)[/tex], we need to evaluate each option for the provided [tex]\( x \)[/tex] values and see which function produces the sequence [tex]\( \{6, 9, 14, 21\} \)[/tex].
Let's evaluate each given function step-by-step:
### Option A: [tex]\( f(x) = 2x + 4 \)[/tex]
1. For [tex]\( x = 1 \)[/tex]:
[tex]\( f(1) = 2(1) + 4 = 2 + 4 = 6 \)[/tex]
2. For [tex]\( x = 2 \)[/tex]:
[tex]\( f(2) = 2(2) + 4 = 4 + 4 = 8 \)[/tex]
3. For [tex]\( x = 3 \)[/tex]:
[tex]\( f(3) = 2(3) + 4 = 6 + 4 = 10 \)[/tex]
4. For [tex]\( x = 4 \)[/tex]:
[tex]\( f(4) = 2(4) + 4 = 8 + 4 = 12 \)[/tex]
The values are [tex]\( \{6, 8, 10, 12\} \)[/tex], which does not match the given sequence.
### Option B: [tex]\( f(x) = x^2 + 5 \)[/tex]
1. For [tex]\( x = 1 \)[/tex]:
[tex]\( f(1) = 1^2 + 5 = 1 + 5 = 6 \)[/tex]
2. For [tex]\( x = 2 \)[/tex]:
[tex]\( f(2) = 2^2 + 5 = 4 + 5 = 9 \)[/tex]
3. For [tex]\( x = 3 \)[/tex]:
[tex]\( f(3) = 3^2 + 5 = 9 + 5 = 14 \)[/tex]
4. For [tex]\( x = 4 \)[/tex]:
[tex]\( f(4) = 4^2 + 5 = 16 + 5 = 21 \)[/tex]
The values are [tex]\( \{6, 9, 14, 21\} \)[/tex], which matches the given sequence.
### Option C: [tex]\( f(x) = x + 5 \)[/tex]
1. For [tex]\( x = 1 \)[/tex]:
[tex]\( f(1) = 1 + 5 = 6 \)[/tex]
2. For [tex]\( x = 2 \)[/tex]:
[tex]\( f(2) = 2 + 5 = 7 \)[/tex]
3. For [tex]\( x = 3 \)[/tex]:
[tex]\( f(3) = 3 + 5 = 8 \)[/tex]
4. For [tex]\( x = 4 \)[/tex]:
[tex]\( f(4) = 4 + 5 = 9 \)[/tex]
The values are [tex]\( \{6, 7, 8, 9\} \)[/tex], which does not match the given sequence.
### Option D: [tex]\( f(x) = 2x^2 + 4 \)[/tex]
1. For [tex]\( x = 1 \)[/tex]:
[tex]\( f(1) = 2(1)^2 + 4 = 2 + 4 = 6 \)[/tex]
2. For [tex]\( x = 2 \)[/tex]:
[tex]\( f(2) = 2(2)^2 + 4 = 8 + 4 = 12 \)[/tex]
3. For [tex]\( x = 3 \)[/tex]:
[tex]\( f(3) = 2(3)^2 + 4 = 18 + 4 = 22 \)[/tex]
4. For [tex]\( x = 4 \)[/tex]:
[tex]\( f(4) = 2(4)^2 + 4 = 32 + 4 = 36 \)[/tex]
The values are [tex]\( \{6, 12, 22, 36\} \)[/tex], which does not match the given sequence.
Upon evaluating all options, we see that only Option B ([tex]\( f(x) = x^2 + 5 \)[/tex]) matches the given sequence [tex]\( \{6, 9, 14, 21\} \)[/tex].
Therefore, the function used to make the pattern is:
[tex]\[ \boxed{f(x) = x^2 + 5} \][/tex]
