Given a linear function f(x)=3x−2f(x) = 3_x - 2f(x)=3x−2, we start by replacing f(x)f(x)f(x) with yyy, resulting in the equation y=3x−2y = 3_x - 2y=3x−2.
To find the inverse function, we swap xxx and yyy in the equation, leading to x=3∗y−2x = 3*y - 2x=3∗y−2.
We isolate yyy on one side of the equation by first subtracting bbb from both sides, resulting in x+2=3∗yx + 2 = 3*yx+2=3∗y. Then, we divide both sides by aaa to get x+23=y\frac{x + 2}{3} = y3x+2=y.
Finally, we replace yyy with f−1(x)f^{-1}(x)f−1(x) to express the inverse function in terms of xxx: f−1(x)=x−baf^{-1}(x) = \frac{x - b}{a}f−1(x)=ax−b, which simplifies to f−1(x)=x/3+2/3f^{-1}(x) = x/3 + 2/3f−1(x)=x/3+2/3.
The inverse function of f(x)=3∗x−2f(x) = 3*x - 2f(x)=3∗x−2 is f−1(x)=x/3+2/3f^{-1}(x) = x/3 + 2/3f−1(x)=x/3+2/3.
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