To find [tex]\( f(6) \)[/tex] for the given piecewise function, we need to determine which part of the function to use based on the value of [tex]\( x = 6 \)[/tex].
The piecewise function is defined as:
[tex]\[
\begin{cases}
y = -2x + 8 & \text{if } x < 7 \\
y = \frac{3}{7}x - 9 & \text{if } x \geq 7
\end{cases}
\][/tex]
Since [tex]\( 6 < 7 \)[/tex], we use the first piece of the function, [tex]\( y = -2x + 8 \)[/tex].
Step-by-step, let's calculate [tex]\( f(6) \)[/tex]:
1. Substitute [tex]\( x = 6 \)[/tex] into the equation [tex]\( y = -2x + 8 \)[/tex]:
[tex]\[
y = -2(6) + 8
\][/tex]
2. Perform the multiplication:
[tex]\[
y = -12 + 8
\][/tex]
3. Perform the addition:
[tex]\[
y = -4
\][/tex]
Thus, [tex]\( f(6) = -4 \)[/tex].
