Let's solve this step-by-step:
1. We are given the functional equation: [tex]\( f(4x - 15) = 8x - 27 \)[/tex].
2. To find the function [tex]\( f(t) \)[/tex] in terms of [tex]\( t \)[/tex], let's make a substitution. Assume [tex]\( t = 4x - 15 \)[/tex].
3. Solve for [tex]\( x \)[/tex] in terms of [tex]\( t \)[/tex]:
[tex]\[
t = 4x - 15 \implies t + 15 = 4x \implies x = \frac{t + 15}{4}
\][/tex]
4. Substitute this [tex]\( x \)[/tex] back into [tex]\( 8x - 27 \)[/tex]:
[tex]\[
f(t) = 8 \left( \frac{t + 15}{4} \right) - 27
\][/tex]
5. Simplifying the equation above:
[tex]\[
f(t) = 2(t + 15) - 27 = 2t + 30 - 27 = 2t + 3
\][/tex]
6. Now that we have [tex]\( f(t) = 2t + 3 \)[/tex], we can find [tex]\( f(f(z)) \)[/tex] for any [tex]\( z \)[/tex].
7. Specifically, we need to find [tex]\( f(f(2)) \)[/tex]:
[tex]\[
\text{First, calculate } f(2):
\][/tex]
[tex]\[
f(2) = 2(2) + 3 = 4 + 3 = 7
\][/tex]
8. Next, use this result to find [tex]\( f(f(2)) \)[/tex]:
[tex]\[
f(f(2)) = f(7)
\][/tex]
[tex]\[
f(7) = 2(7) + 3 = 14 + 3 = 17
\][/tex]
Hence, the value of [tex]\( f(f(2)) \)[/tex] is [tex]\( \boxed{17} \)[/tex].
