To find the inverse function [tex]\( g^{-1}(x) \)[/tex] of the given function [tex]\( g(x) = \frac{5x - 3}{2x + 1} \)[/tex], and then evaluate [tex]\( g^{-1}(2) \)[/tex], follow these steps:
### Step 1: Express the Function in Terms of [tex]\( y \)[/tex]
First, we start by replacing [tex]\( g(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[ y = \frac{5x - 3}{2x + 1} \][/tex]
### Step 2: Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to Find the Inverse
Next, we swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to solve for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[ x = \frac{5y - 3}{2y + 1} \][/tex]
### Step 3: Solve for [tex]\( y \)[/tex]
To isolate [tex]\( y \)[/tex], we need to clear the fraction by multiplying both sides by [tex]\( 2y + 1 \)[/tex]:
[tex]\[ x(2y + 1) = 5y - 3 \][/tex]
Distribute [tex]\( x \)[/tex]:
[tex]\[ 2xy + x = 5y - 3 \][/tex]
Rearrange the equation to get all terms involving [tex]\( y \)[/tex] on one side and constants on the other:
[tex]\[ 2xy - 5y = -x - 3 \][/tex]
Factor out [tex]\( y \)[/tex]:
[tex]\[ y(2x - 5) = -x - 3 \][/tex]
Finally, solve for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{-x - 3}{2x - 5} \][/tex]
So, the inverse function is:
[tex]\[ g^{-1}(x) = \frac{-x - 3}{2x - 5} \][/tex]
### Step 4: Evaluate [tex]\( g^{-1}(2) \)[/tex]
To find [tex]\( g^{-1}(2) \)[/tex], substitute [tex]\( x = 2 \)[/tex] into [tex]\( g^{-1}(x) \)[/tex]:
[tex]\[ g^{-1}(2) = \frac{-2 - 3}{2(2) - 5} \][/tex]
Calculate the numerator and the denominator separately:
[tex]\[ \text{Numerator: } -2 - 3 = -5 \][/tex]
[tex]\[ \text{Denominator: } 2(2) - 5 = 4 - 5 = -1 \][/tex]
Now, divide the numerator by the denominator:
[tex]\[ g^{-1}(2) = \frac{-5}{-1} = 5 \][/tex]
### Summary of Results
1. The inverse function is:
[tex]\[ g^{-1}(x) = \frac{-x - 3}{2x - 5} \][/tex]
2. The value of the inverse function at [tex]\( x = 2 \)[/tex] is:
[tex]\[ g^{-1}(2) = 5 \][/tex]
So, the inverse function [tex]\( g^{-1}(x) \)[/tex] and its value at [tex]\( 2 \)[/tex] are correctly determined as:
[tex]\[ g^{-1}(x) = \frac{-x - 3}{2x - 5} \][/tex]
[tex]\[ g^{-1}(2) = 5 \][/tex]
