Certainly! Let's solve the given problem step-by-step.
Given:
1. [tex]\( f(x) = 2x - 3 \)[/tex]
2. [tex]\( g(x) = \frac{2x - 7}{3} \)[/tex]
We are given the equation [tex]\( f(f(x)) = g^{-1}(x) \)[/tex] and need to find the value of [tex]\( x \)[/tex].
### Step 1: Calculate [tex]\( f(f(x)) \)[/tex]
First, we need to find [tex]\( f(f(x)) \)[/tex]:
[tex]\[ f(f(x)) = f(2x - 3) \][/tex]
To do this, we substitute [tex]\( 2x - 3 \)[/tex] into the function [tex]\( f \)[/tex] again:
[tex]\[ f(2x - 3) = 2(2x - 3) - 3 \][/tex]
[tex]\[ = 4x - 6 - 3 \][/tex]
[tex]\[ = 4x - 9 \][/tex]
Thus, [tex]\( f(f(x)) = 4x - 9 \)[/tex].
### Step 2: Find the inverse of [tex]\( g(x) \)[/tex]
Next, we need to find the inverse of [tex]\( g(x) \)[/tex]. Let [tex]\( y = g(x) \)[/tex]:
[tex]\[ y = \frac{2x - 7}{3} \][/tex]
To find the inverse, we solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[ y = \frac{2x - 7}{3} \][/tex]
Multiply both sides by 3:
[tex]\[ 3y = 2x - 7 \][/tex]
Add 7 to both sides:
[tex]\[ 3y + 7 = 2x \][/tex]
Divide both sides by 2:
[tex]\[ x = \frac{3y + 7}{2} \][/tex]
Thus, the inverse function [tex]\( g^{-1}(x) \)[/tex] is:
[tex]\[ g^{-1}(x) = \frac{3x + 7}{2} \][/tex]
### Step 3: Set up the equation [tex]\( f(f(x)) = g^{-1}(x) \)[/tex]
We have [tex]\( f(f(x)) = 4x - 9 \)[/tex] and [tex]\( g^{-1}(x) = \frac{3x + 7}{2} \)[/tex].
Set them equal to each other:
[tex]\[ 4x - 9 = \frac{3x + 7}{2} \][/tex]
### Step 4: Solve for [tex]\( x \)[/tex]
To eliminate the fraction, multiply both sides by 2:
[tex]\[ 2(4x - 9) = 3x + 7 \][/tex]
[tex]\[ 8x - 18 = 3x + 7 \][/tex]
Subtract [tex]\( 3x \)[/tex] from both sides:
[tex]\[ 8x - 3x - 18 = 7 \][/tex]
[tex]\[ 5x - 18 = 7 \][/tex]
Add 18 to both sides:
[tex]\[ 5x = 25 \][/tex]
Divide by 5:
[tex]\[ x = 5 \][/tex]
### Conclusion
The solution to the given equation [tex]\( f(f(x)) = g^{-1}(x) \)[/tex] is:
[tex]\[ \boxed{5} \][/tex]
