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Hello!

$\large\boxed{f^{-1} = \frac{5x+2}{4x-2}, f^{-1}(3) = \frac{17}{10} }$

$f(x) = \frac{2x +2}{4x - 5}$

Find the inverse by swapping the x and y variables:

$y = \frac{2x +2}{4x - 5}\\\\x = \frac{2y +2}{4y - 5}$

Begin simplifying. Multiply both sides by 4y - 5:

$x(4y - 5) = 2y + 2$

Start isolating for y by subtracting 2 from both sides:

$x(4y - 5) -2 = 2y$

Distribute x:

$4yx - 5x - 2 = 2y$

Move the term involving y (4yx) over to the other side:

$- 5x - 2 = 2y - 4yx\\\\$

Factor out y and divide:

$- 5x - 2 = y(2 - 4x)\\\\y = \frac{-5x-2}{-4x+2} \\\\y = \frac{-(5x + 2)}{-(4x - 2)} \\\\y^{-1} = \frac{5x + 2}{4x - 2}$

Use this equation to evaluate $f^{-1}(3)$

$f^{-1}(3) = \frac{5(3) + 2}{4(3) - 2} = \frac{17}{10}$

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answered 9 months ago