Sure, let's address the given problem step by step.
### Part (a): Find the inverse of [tex]\( g(x) \)[/tex]
First, we start with the function [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = \frac{\sqrt{x} + 2}{5} \][/tex]
To find the inverse [tex]\( g^{-1}(x) \)[/tex], follow these steps:
1. Replace [tex]\( g(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[ y = \frac{\sqrt{x} + 2}{5} \][/tex]
2. Solve this equation for [tex]\( x \)[/tex]:
[tex]\[ y \cdot 5 = \sqrt{x} + 2 \][/tex]
[tex]\[ 5y = \sqrt{x} + 2 \][/tex]
[tex]\[ 5y - 2 = \sqrt{x} \][/tex]
[tex]\[ x = (5y - 2)^2 \][/tex]
Therefore, the inverse function [tex]\( g^{-1}(x) \)[/tex] is:
[tex]\[ g^{-1}(x) = (5x - 2)^2 \][/tex]
### Part (b): Solve [tex]\( gf(x) = 3 \)[/tex]
Next, we need to solve the equation [tex]\( gf(x) = 3 \)[/tex], where [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are given as:
[tex]\[ f(x) = 3x + 4 \][/tex]
[tex]\[ g(f(x)) = \frac{\sqrt{f(x)} + 2}{5} \][/tex]
So, the combined function [tex]\( gf(x) \)[/tex] is:
[tex]\[ g(f(x)) = \frac{\sqrt{3x + 4}+2}{5} \][/tex]
Set this equal to [tex]\( 3 \)[/tex] and solve for [tex]\( x \)[/tex]:
[tex]\[ \frac{\sqrt{3x + 4} + 2}{5} = 3 \][/tex]
1. Eliminate the fraction by multiplying both sides by 5:
[tex]\[ \sqrt{3x + 4} + 2 = 15 \][/tex]
2. Isolate the square root term:
[tex]\[ \sqrt{3x + 4} = 13 \][/tex]
3. Square both sides to remove the square root:
[tex]\[ 3x + 4 = 169 \][/tex]
4. Solve for [tex]\( x \)[/tex] by isolating it on one side:
[tex]\[ 3x = 165 \][/tex]
[tex]\[ x = 55 \][/tex]
So, the solution to the equation [tex]\( gf(x) = 3 \)[/tex] is:
[tex]\[ x = 55 \][/tex]
### Summary
(a) The inverse function is:
[tex]\[ g^{-1}(x) = (5x - 2)^2 \][/tex]
(b) The solution to [tex]\( gf(x) = 3 \)[/tex] is:
[tex]\[ x = 55 \][/tex]
