Inverse of a Function

Calculus Level 2

If f ( x ) = 2 x − 3 3 x − 2 f(x) = \dfrac {2x-3}{3x-2} , find its inverse f − 1 ( x ) f^{-1}(x) .

2 x + 3 3 x − 2 \frac {2x+3}{3x-2} 2 x − 3 3 x + 2 \frac {2x-3}{3x+2} 2 − 3 x 3 − 2 x \frac {2-3x}{3-2x} 2 x − 3 3 x − 2 \frac {2x-3}{3x-2}

César Emanuel Castro by César Emanuel Castro , 20, Honduras

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1 solution

Chew-Seong Cheong
Apr 15, 2018

Let y = f ( x ) ⟹ f − 1 ( y ) = x y = f(x)\implies f^{-1}(y) = x .

f ( x ) = 2 x − 3 3 x − 2 y = 2 x − 3 3 x − 2 3 x y − 2 y = 2 x − 3 3 x y − 2 x = 2 y − 3 x = 2 y − 3 3 y − 3 f − 1 ( y ) = 2 y − 3 3 y − 3 Replace y with x . f − 1 ( x ) = 2 x − 3 3 x − 3 \begin{aligned} f(x) & = \frac {2x-3}{3x-2} \\ y & = \frac {2x-3}{3x-2} \\ 3xy-2y & = 2x-3 \\ 3xy-2x & = 2y-3 \\ x & = \frac {2y-3}{3y-3} \\ f^{-1}(y) & = \frac {2y-3}{3y-3} & \small \color{#3D99F6} \text{Replace }y \text{ with }x. \\ f^{-1}(x) & = \boxed{\dfrac {2x-3}{3x-3}} \end{aligned}

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