To find the value of [tex]\( t(6) \)[/tex] in the given piecewise function, we start by identifying which part of the function to use based on the value of [tex]\( x = 6 \)[/tex].
The function is defined as:
[tex]\[
y =
\begin{cases}
-3x + 19 & \text{if } x < 4 \\
\frac{x}{2} + 5 & \text{if } x \geq 4
\end{cases}
\][/tex]
Since [tex]\( x = 6 \)[/tex] is greater than 4, we use the second part of the function:
[tex]\[
y = \frac{x}{2} + 5
\][/tex]
Now we substitute [tex]\( x = 6 \)[/tex] into this equation:
[tex]\[
y = \frac{6}{2} + 5
\][/tex]
First, we perform the division:
[tex]\[
\frac{6}{2} = 3
\][/tex]
Next, we add 5 to the result of the division:
[tex]\[
3 + 5 = 8
\][/tex]
Therefore, the value of [tex]\( t(6) \)[/tex] is [tex]\( 8 \)[/tex].
So, [tex]\( t(6) = 8 \)[/tex].
