To determine the equation of the inverse of the function [tex]\( y = 4^{2x + 9} \)[/tex], we need to follow a series of steps to solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex] and then interchange [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
1. Rewrite the given function:
[tex]\[
y = 4^{2x + 9}
\][/tex]
2. Switch [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to find the inverse function:
[tex]\[
x = 4^{2y + 9}
\][/tex]
3. Take the logarithm base 4 of both sides to solve for [tex]\( y \)[/tex]:
Recall that if [tex]\( a = b^c \)[/tex], then [tex]\( \log_b(a) = c \)[/tex].
[tex]\[
\log_4(x) = 2y + 9
\][/tex]
4. Isolate [tex]\( y \)[/tex]:
[tex]\[
\log_4(x) - 9 = 2y
\][/tex]
[tex]\[
y = \frac{\log_4(x) - 9}{2}
\][/tex]
Thus, the inverse function of [tex]\( y = 4^{2x + 9} \)[/tex] is:
[tex]\[
y = \frac{\log_4(x) - 9}{2}
\][/tex]
Therefore, the correct answer is:
[tex]\[
y = \frac{\log_4(x) - 9}{2}
\][/tex]
