To find the inverse of the function [tex]\( f(x) = 2^x + 6 \)[/tex], we need to follow these steps:
1. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[
y = 2^x + 6
\][/tex]
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to start finding the inverse:
[tex]\[
x = 2^y + 6
\][/tex]
3. Solve for [tex]\( y \)[/tex]:
- First, isolate the exponential term:
[tex]\[
x - 6 = 2^y
\][/tex]
- Next, take the logarithm base 2 of both sides to solve for [tex]\( y \)[/tex]:
[tex]\[
\log_2(x - 6) = y
\][/tex]
4. Thus, the inverse function [tex]\( f^{-1}(x) \)[/tex] is:
[tex]\[
f^{-1}(x) = \log_2(x - 6)
\][/tex]
Therefore, the correct option is:
D. [tex]\( f^{-1}(x) = \log_2(x - 6) \)[/tex]
