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Answer :
To evaluate the function [tex]\( f(x) \)[/tex] when [tex]\( x = 6 \)[/tex], we look at the piecewise definition of the function:
1. First part of the function: [tex]\( f(x) = 6x^2 + 2 \)[/tex] if [tex]\(-6 < x < 9\)[/tex].
2. Second part of the function: [tex]\( f(x) = 12 \)[/tex] if [tex]\( 9 \leq x < 13\)[/tex].
Since [tex]\( x = 6 \)[/tex] falls within the range [tex]\(-6 < x < 9\)[/tex], the first part of the function applies. Therefore, we use the equation [tex]\( f(x) = 6x^2 + 2 \)[/tex].
Let's calculate [tex]\( f(6) \)[/tex]:
- Substitute [tex]\( x = 6 \)[/tex] into [tex]\( 6x^2 + 2 \)[/tex]:
[tex]\[
f(6) = 6 \times 6^2 + 2
\][/tex]
- Calculate [tex]\( 6^2 \)[/tex] first:
[tex]\[
6^2 = 36
\][/tex]
- Multiply by 6:
[tex]\[
6 \times 36 = 216
\][/tex]
- Add 2:
[tex]\[
216 + 2 = 218
\][/tex]
Therefore, the value of [tex]\( f(6) \)[/tex] is 218.
1. First part of the function: [tex]\( f(x) = 6x^2 + 2 \)[/tex] if [tex]\(-6 < x < 9\)[/tex].
2. Second part of the function: [tex]\( f(x) = 12 \)[/tex] if [tex]\( 9 \leq x < 13\)[/tex].
Since [tex]\( x = 6 \)[/tex] falls within the range [tex]\(-6 < x < 9\)[/tex], the first part of the function applies. Therefore, we use the equation [tex]\( f(x) = 6x^2 + 2 \)[/tex].
Let's calculate [tex]\( f(6) \)[/tex]:
- Substitute [tex]\( x = 6 \)[/tex] into [tex]\( 6x^2 + 2 \)[/tex]:
[tex]\[
f(6) = 6 \times 6^2 + 2
\][/tex]
- Calculate [tex]\( 6^2 \)[/tex] first:
[tex]\[
6^2 = 36
\][/tex]
- Multiply by 6:
[tex]\[
6 \times 36 = 216
\][/tex]
- Add 2:
[tex]\[
216 + 2 = 218
\][/tex]
Therefore, the value of [tex]\( f(6) \)[/tex] is 218.
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