For the function f, of, x, equals, start fraction, 3, x, minus, 5, divided by, 3, minus, 4, x, end fractionf(x)= 3−4x3x−5 , find f, to the power minus 1 , left parenthesis, x, right parenthesisf −1 (x).
Question
For the function f, of, x, equals, start fraction, 3, x, minus, 5, divided by, 3, minus, 4, x, end fractionf(x)= 3−4x3x−5 , find f, to the power minus 1 , left parenthesis, x, right parenthesisf −1 (x).
Solution
To find the inverse of the function f(x) = (3x - 5) / (3 - 4x), follow these steps:
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Replace f(x) with y. This gives us y = (3x - 5) / (3 - 4x).
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Swap x and y. This gives us x = (3y - 5) / (3 - 4y).
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Solve the equation for y.
First, cross-multiply to get rid of the fraction: x(3 - 4y) = 3y - 5.
This simplifies to 3x - 4xy = 3y - 5.
Rearrange the terms to group all terms involving y on one side of the equation and the constants on the other side: 3x + 5 = 3y + 4xy.
Factor out y: 3x + 5 = y(3 + 4x).
Finally, solve for y by dividing both sides by (3 + 4x): y = (3x + 5) / (3 + 4x).
So, the inverse of the function f, denoted by f^(-1)(x), is f^(-1)(x) = (3x + 5) / (3 + 4x).
Similar Questions
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For the function f, of, x, equals, start fraction, 8, divided by, 9, plus, 4, x, end fractionf(x)= 9+4x8 , find f, to the power minus 1 , left parenthesis, x, right parenthesisf −1 (x).
For the function f, of, x, equals, left parenthesis, 6, x, right parenthesis, start superscript, one third, end superscriptf(x)=(6x) 31 , find f, to the power minus 1 , left parenthesis, x, right parenthesisf −1 (x).
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