Let's solve this step-by-step.
We are given two functions:
[tex]\[ f(x) = 2x + 1 \][/tex]
[tex]\[ g(x) = \frac{3x + 1}{2} \][/tex]
We are also given that [tex]\( f(x) = g^{-1}(x) \)[/tex], where [tex]\( g^{-1}(x) \)[/tex] is the inverse function of [tex]\( g(x) \)[/tex]. To find the value of [tex]\( x \)[/tex], we need to follow these steps:
### Step 1: Find the Inverse of [tex]\( g(x) \)[/tex]
To find the inverse of [tex]\( g(x) \)[/tex], we first set [tex]\( y = g(x) \)[/tex]:
[tex]\[ y = \frac{3x + 1}{2} \][/tex]
Next, we solve this equation for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
1. Multiply both sides by 2 to clear the denominator:
[tex]\[ 2y = 3x + 1 \][/tex]
2. Subtract 1 from both sides:
[tex]\[ 2y - 1 = 3x \][/tex]
3. Divide by 3 to solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{2y - 1}{3} \][/tex]
So, the inverse function [tex]\( g^{-1}(x) \)[/tex] is:
[tex]\[ g^{-1}(x) = \frac{2x - 1}{3} \][/tex]
### Step 2: Set [tex]\( f(x) \)[/tex] Equal to [tex]\( g^{-1}(x) \)[/tex]
We know that:
[tex]\[ f(x) = g^{-1}(x) \][/tex]
Substitute the expressions for [tex]\( f(x) \)[/tex] and [tex]\( g^{-1}(x) \)[/tex]:
[tex]\[ 2x + 1 = \frac{2x - 1}{3} \][/tex]
### Step 3: Solve for [tex]\( x \)[/tex]
Now, solve the equation:
1. Clear the fraction by multiplying both sides by 3:
[tex]\[ 3(2x + 1) = 2x - 1 \][/tex]
2. Distribute the 3 on the left side:
[tex]\[ 6x + 3 = 2x - 1 \][/tex]
3. Subtract 2x from both sides to collect [tex]\( x \)[/tex] terms on one side:
[tex]\[ 6x - 2x + 3 = -1 \][/tex]
[tex]\[ 4x + 3 = -1 \][/tex]
4. Subtract 3 from both sides:
[tex]\[ 4x = -1 - 3 \][/tex]
[tex]\[ 4x = -4 \][/tex]
5. Divide by 4:
[tex]\[ x = -1 \][/tex]
So, the value of [tex]\( x \)[/tex] that satisfies the given condition is:
[tex]\[ x = -1 \][/tex]
