Answer :
To evaluate [tex]\( f(x) \)[/tex] when [tex]\( x = 6 \)[/tex], we need to determine which part of the piecewise function applies to our value of [tex]\( x \)[/tex].
Here is the piecewise function:
[tex]\[
f(x) =
\begin{cases}
3x^2 + 1 & \text{if } -4 < x < 6 \\
6 & \text{if } 6 \leq x < 9
\end{cases}
\][/tex]
Now let's look at the conditions for [tex]\( x = 6 \)[/tex]:
1. The first condition, [tex]\( -4 < x < 6 \)[/tex], does not include [tex]\( x = 6 \)[/tex] because it is strictly less than 6. Therefore, we cannot use this part of the function.
2. The second condition, [tex]\( 6 \leq x < 9 \)[/tex], includes [tex]\( x = 6 \)[/tex] since it's defined for values of [tex]\( x \)[/tex] that are greater than or equal to 6. Therefore, this is the appropriate condition to use for [tex]\( x = 6 \)[/tex].
According to the second part of the function, when [tex]\( 6 \leq x < 9 \)[/tex], [tex]\( f(x) = 6 \)[/tex].
Thus, for [tex]\( x = 6 \)[/tex], the value of the function [tex]\( f(x) \)[/tex] is [tex]\( 6 \)[/tex].
Here is the piecewise function:
[tex]\[
f(x) =
\begin{cases}
3x^2 + 1 & \text{if } -4 < x < 6 \\
6 & \text{if } 6 \leq x < 9
\end{cases}
\][/tex]
Now let's look at the conditions for [tex]\( x = 6 \)[/tex]:
1. The first condition, [tex]\( -4 < x < 6 \)[/tex], does not include [tex]\( x = 6 \)[/tex] because it is strictly less than 6. Therefore, we cannot use this part of the function.
2. The second condition, [tex]\( 6 \leq x < 9 \)[/tex], includes [tex]\( x = 6 \)[/tex] since it's defined for values of [tex]\( x \)[/tex] that are greater than or equal to 6. Therefore, this is the appropriate condition to use for [tex]\( x = 6 \)[/tex].
According to the second part of the function, when [tex]\( 6 \leq x < 9 \)[/tex], [tex]\( f(x) = 6 \)[/tex].
Thus, for [tex]\( x = 6 \)[/tex], the value of the function [tex]\( f(x) \)[/tex] is [tex]\( 6 \)[/tex].
