For [tex]\( x \geqslant 0 \)[/tex], the functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are such that:
[tex]\[
f(x) = 3x + 4 \quad \text{and} \quad g(x) = \frac{\sqrt{x} + 2}{5}
\][/tex]

(a) Find [tex]\( g^{-1}(x) \)[/tex]
[tex]\[
g^{-1}(x) =
\][/tex]

(b) Solve [tex]\( \operatorname{gf}(x) = 3 \)[/tex]
[tex]\[
\operatorname{gf}(x) = 3
\][/tex]



Answer :

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]