To find the inverse function [tex]\( f^{-1}(x) \)[/tex] for the function [tex]\( f(x) = 3x - 2 \)[/tex], follow these steps:
1. Express [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
Start by setting [tex]\( f(x) = y \)[/tex], which means:
[tex]\[ y = 3x - 2. \][/tex]
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
In the inverse function, the roles of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] are reversed, so swap them to rewrite the equation:
[tex]\[ x = 3y - 2. \][/tex]
3. Solve the equation for [tex]\( y \)[/tex]:
Isolate [tex]\( y \)[/tex] on one side of the equation to express it in terms of [tex]\( x \)[/tex]. Add [tex]\( 2 \)[/tex] to both sides:
[tex]\[ x + 2 = 3y. \][/tex]
Then, divide both sides by [tex]\( 3 \)[/tex] to solve for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{x + 2}{3}. \][/tex]
4. Write the inverse function:
The expression [tex]\( y = \frac{x + 2}{3} \)[/tex] represents the inverse function [tex]\( f^{-1}(x) \)[/tex]. Hence,
[tex]\[ f^{-1}(x) = \frac{x + 2}{3}. \][/tex]
So, the inverse function [tex]\( f^{-1}(x) \)[/tex] of the given function [tex]\( f(x) = 3x - 2 \)[/tex] is:
[tex]\[ f^{-1}(x) = \frac{x + 2}{3}. \][/tex]
