To find the inverse of the function [tex]\( y = 3x - 2 \)[/tex], we will follow these steps:
1. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
The idea of finding an inverse function is to find a function that will reverse the effect of the original function. To do this, we swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
Starting with [tex]\( y = 3x - 2 \)[/tex]:
[tex]\[
x = 3y - 2
\][/tex]
2. Solve for [tex]\( y \)[/tex]:
We need to isolate [tex]\( y \)[/tex] to find the inverse function.
- First, add 2 to both sides:
[tex]\[
x + 2 = 3y
\][/tex]
- Next, divide both sides by 3:
[tex]\[
y = \frac{x + 2}{3}
\][/tex]
3. Write the inverse function:
The inverse function is [tex]\( y^{-1}(x) = \frac{x + 2}{3} \)[/tex].
Now, we need to compare this result with the given options:
- [tex]\( y^{-1} = -\frac{2}{3}x + \frac{2}{3} \)[/tex]
- [tex]\( y^{-1} = \frac{2}{3}x - \frac{2}{3} \)[/tex]
- [tex]\( y^{-1} = \frac{1}{3}x + \frac{2}{3} \)[/tex]
- [tex]\( y^{-1} = -\frac{1}{3}x - \frac{2}{3} \)[/tex]
The correct inverse function we derived is:
[tex]\[
y^{-1}(x) = \frac{1}{3} x + \frac{2}{3}
\][/tex]
Therefore, the correct option is:
[tex]\[
y^{-1} = \frac{1}{3} x + \frac{2}{3}
\][/tex]
