Sure, let's work through each part of the problem step-by-step.
### (a) Finding [tex]\( g^{-1}(x) \)[/tex]
The function [tex]\( g(x) \)[/tex] is given by:
[tex]\[ g(x) = 3x + 1 \][/tex]
To find the inverse function [tex]\( g^{-1}(x) \)[/tex], we need to solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex] where [tex]\( y = g(x) \)[/tex].
Step 1: Set [tex]\( y = g(x) \)[/tex]:
[tex]\[ y = 3x + 1 \][/tex]
Step 2: Solve for [tex]\( x \)[/tex]:
[tex]\[ y - 1 = 3x \][/tex]
[tex]\[ x = \frac{y - 1}{3} \][/tex]
Step 3: Replace [tex]\( y \)[/tex] with [tex]\( x \)[/tex] to get the inverse function:
[tex]\[ g^{-1}(x) = \frac{x - 1}{3} \][/tex]
### (b) Finding the composition [tex]\( \operatorname{gh}(x) \)[/tex]
The function [tex]\( h(x) \)[/tex] is given by:
[tex]\[ h(x) = \frac{x - 1}{2} \][/tex]
The composition [tex]\( \operatorname{gh}(x) \)[/tex] means we need to substitute [tex]\( h(x) \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[ \operatorname{gh}(x) = g(h(x)) \][/tex]
Step 1: Substitute [tex]\( h(x) \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[ g\left( \frac{x - 1}{2} \right) \][/tex]
Step 2: Calculate [tex]\( g\left( \frac{x - 1}{2} \right) \)[/tex]:
[tex]\[ g\left( \frac{x - 1}{2} \right) = 3\left(\frac{x - 1}{2}\right) + 1 \][/tex]
[tex]\[ = \frac{3(x - 1)}{2} + 1 \][/tex]
[tex]\[ = \frac{3x - 3}{2} + \frac{2}{2} \][/tex]
[tex]\[ = \frac{3x - 3 + 2}{2} \][/tex]
[tex]\[ = \frac{3x - 1}{2} \][/tex]
Therefore,
[tex]\[ \operatorname{gh}(x) = \frac{3x - 1}{2} \][/tex]
### (c) Finding [tex]\( x \)[/tex] if [tex]\( \operatorname{gg}(x) = 22 \)[/tex]
We need to find [tex]\( x \)[/tex] such that [tex]\( gg(x) = 22 \)[/tex]. This means we need to substitute [tex]\( g(x) \)[/tex] into itself:
[tex]\[ gg(x) = g(g(x)) \][/tex]
Step 1: Calculate [tex]\( g(g(x)) \)[/tex]:
[tex]\[ g(g(x)) = g(3x + 1) \][/tex]
[tex]\[ = 3(3x + 1) + 1 \][/tex]
[tex]\[ = 9x + 3 + 1 \][/tex]
[tex]\[ = 9x + 4 \][/tex]
Step 2: Set this equal to 22 and solve for [tex]\( x \)[/tex]:
[tex]\[ 9x + 4 = 22 \][/tex]
[tex]\[ 9x = 22 - 4 \][/tex]
[tex]\[ 9x = 18 \][/tex]
[tex]\[ x = \frac{18}{9} \][/tex]
[tex]\[ x = 2 \][/tex]
Therefore, the solutions are:
(a) [tex]\( g^{-1}(x) = \frac{x - 1}{3} \)[/tex]
(b) [tex]\( \operatorname{gh}(x) = \frac{3x - 1}{2} \)[/tex]
(c) [tex]\( x = 2 \)[/tex] if [tex]\( \operatorname{gg}(x) = 22 \)[/tex]
