For the function f, of, x, equals, start fraction, x, divided by, 2, x, plus, 3, end fractionf(x)= 2x+3x , find f, to the power minus 1 , left parenthesis, x, right parenthesisf −1 (x).
Question
For the function f, of, x, equals, start fraction, x, divided by, 2, x, plus, 3, end fractionf(x)= 2x+3x , find f, to the power minus 1 , left parenthesis, x, right parenthesisf −1 (x).
Solution
To find the inverse of the function f(x) = x/(2x + 3), follow these steps:
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Replace f(x) with y, so the equation becomes y = x/(2x + 3).
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Swap x and y to get x = y/(2y + 3).
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Solve the equation for y to find the inverse function.
Multiply both sides by (2y + 3) to get rid of the denominator on the right side: x(2y + 3) = y.
Distribute x on the left side: 2xy + 3x = y.
Move the term with y on the left side to the right side: 2xy - y = -3x.
Factor out y on the right side: y(2x - 1) = -3x.
Divide both sides by (2x - 1) to solve for y: y = -3x/(2x - 1).
So, the inverse function f^(-1)(x) is f^(-1)(x) = -3x/(2x - 1).
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