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What is the inverse of [tex]y = 3x - 2[/tex]?

A. [tex]y^{-1} = -\frac{2}{3}x + \frac{2}{3}[/tex]
B. [tex]y^{-1} = \frac{2}{3}x - \frac{2}{3}[/tex]
C. [tex]y^{-1} = \frac{1}{3}x + \frac{2}{3}[/tex]
D. [tex]y^{-1} = -\frac{1}{3}x - \frac{2}{3}[/tex]


Sagot :

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]