For the function f, of, x, equals, start fraction, 5, plus, 3, x, divided by, 2, minus, 3, x, end fractionf(x)= 2−3x5+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, 5, plus, 3, x, divided by, 2, minus, 3, x, end fractionf(x)= 2−3x5+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) = (5 + 3x) / (2 - 3x), we follow these steps:
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Replace f(x) with y. This gives us y = (5 + 3x) / (2 - 3x).
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Swap x and y. This gives us x = (5 + 3y) / (2 - 3y).
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Solve the equation for y.
First, cross-multiply to get rid of the fraction: x(2 - 3y) = 5 + 3y.
This simplifies to 2x - 3xy = 5 + 3y.
Rearrange the terms to group all terms involving y on one side of the equation: 2x - 5 = 3y + 3xy.
Factor out y: 2x - 5 = y(3 + 3x).
Finally, solve for y: y = (2x - 5) / (3 + 3x).
So, the inverse function, f^(-1)(x), is f^(-1)(x) = (2x - 5) / (3 + 3x).
Similar Questions
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If f, left bracket, x, right bracket, equals, 2, x, to the power 5 , plus, 5, x, minus, 1f(x)=2x 5 +5x−1, then what is the remainder when f, left bracket, x, right bracketf(x) is divided by x, plus, 3x+3?
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