Let's find the inverse of the function [tex]\( g(x) = \frac{5}{x-3} + 6 \)[/tex] step-by-step.
1. Start with the original function:
[tex]\[
g(x) = \frac{5}{x-3} + 6
\][/tex]
2. Replace [tex]\( g(x) \)[/tex] with [tex]\( y \)[/tex] to facilitate finding the inverse:
[tex]\[
y = \frac{5}{x-3} + 6
\][/tex]
3. Interchange [tex]\( x \)[/tex] and [tex]\( y \)[/tex] (this is effectively saying to solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]):
[tex]\[
x = \frac{5}{y-3} + 6
\][/tex]
4. Isolate the fraction by subtracting 6 from both sides:
[tex]\[
x - 6 = \frac{5}{y-3}
\][/tex]
5. To isolate [tex]\( y \)[/tex], multiply both sides by [tex]\( y-3 \)[/tex]:
[tex]\[
(x - 6)(y - 3) = 5
\][/tex]
6. Distribute [tex]\( x - 6 \)[/tex] on the left-hand side:
[tex]\[
xy - 3x - 6y + 18 = 5
\][/tex]
7. Rearrange the terms to isolate those containing [tex]\( y \)[/tex] on one side:
[tex]\[
xy - 6y = 3x - 13
\][/tex]
8. Factor out [tex]\( y \)[/tex] on the left-hand side:
[tex]\[
y(x - 6) = 3x - 13
\][/tex]
9. Solve for [tex]\( y \)[/tex] by dividing both sides by [tex]\( x - 6 \)[/tex]:
[tex]\[
y = \frac{3x - 13}{x - 6}
\][/tex]
So, the inverse of the function [tex]\( g(x) = \frac{5}{x-3} + 6 \)[/tex] is:
[tex]\[
g^{-1}(x) = \frac{3x - 13}{x - 6}
\][/tex]
This is in line with the given solution.
