For the function f, of, x, equals, start fraction, 2, x, plus, 9, divided by, 2, x, minus, 7, end fractionf(x)= 2x−72x+9 , find f, to the power minus 1 , left bracket, x, right bracketf −1 (x).
Question
For the function f, of, x, equals, start fraction, 2, x, plus, 9, divided by, 2, x, minus, 7, end fractionf(x)= 2x−72x+9 , find f, to the power minus 1 , left bracket, x, right bracketf −1 (x).
Solution
To find the inverse of the function f(x) = (2x + 9) / (2x - 7), we follow these steps:
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Replace f(x) with y. This gives us y = (2x + 9) / (2x - 7).
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Swap x and y. This gives us x = (2y + 9) / (2y - 7).
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Solve for y.
First, cross-multiply to get rid of the fraction: x * (2y - 7) = 2y + 9.
This simplifies to 2xy - 7x = 2y + 9.
Rearrange the equation to group the y terms on one side and the constants on the other: 2xy - 2y = 7x + 9.
Factor out y: y * (2x - 2) = 7x + 9.
Finally, solve for y: y = (7x + 9) / (2x - 2).
So, the inverse of the function f, denoted as f^(-1)(x), is f^(-1)(x) = (7x + 9) / (2x - 2).
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