To find the inverse of the function [tex]\( f(x) = 6 + \sqrt{3x - 9} \)[/tex], let's follow these steps:
1. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[
y = 6 + \sqrt{3x - 9}
\][/tex]
2. Solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
- First, isolate the square root term:
[tex]\[
y - 6 = \sqrt{3x - 9}
\][/tex]
- Next, square both sides to eliminate the square root:
[tex]\[
(y - 6)^2 = 3x - 9
\][/tex]
- Now, solve for [tex]\( x \)[/tex]:
[tex]\[
(y - 6)^2 = 3x - 9
\][/tex]
[tex]\[
(y - 6)^2 + 9 = 3x
\][/tex]
[tex]\[
x = \frac{(y - 6)^2 + 9}{3}
\][/tex]
3. Simplify the expression:
[tex]\[
x = \frac{(y - 6)^2 + 9}{3}
\][/tex]
[tex]\[
x = \frac{(y - 6)^2}{3} + \frac{9}{3}
\][/tex]
[tex]\[
x = \frac{(y - 6)^2}{3} + 3
\][/tex]
Thus, the inverse function [tex]\( f^{-1}(x) \)[/tex] is:
[tex]\[
f^{-1}(x) = \frac{(x - 6)^2}{3} + 3
\][/tex]
