To find the inverse of a rational function, you essentially swap the roles of x and y and then solve for y. This process is illustrated in the provided reference.
Steps to Find the Inverse of a Rational Function
- Replace f(x) with y. This is simply notational change to make the subsequent steps clearer.
- Swap x and y. Wherever you see an x, replace it with y, and vice versa. This reflects the fundamental concept of an inverse function: reversing the input and output. According to the video ([1:06], [4:16]), the relationship between x and y will be swapped.
- Solve for y. This is the algebraic manipulation step. Isolate y on one side of the equation. The resulting equation will be in the form y = some expression involving x.
- Replace y with f-1(x). This is the standard notation for the inverse function.
Example
Let's consider a rational function and find its inverse, based on the principles described in the reference.
Suppose we have the function:
f(x) = (2x + 5) / (4 - 3x)
Here's how to find its inverse:
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Replace f(x) with y:
y = (2x + 5) / (4 - 3x)
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Swap x and y:
x = (2y + 5) / (4 - 3y)
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Solve for y:
- Multiply both sides by (4 - 3y): x(4 - 3y) = 2y + 5
- Distribute: 4x - 3xy = 2y + 5
- Gather terms with y on one side: 4x - 5 = 2y + 3xy
- Factor out y: 4x - 5 = y(2 + 3x)
- Divide to isolate y: y = (4x - 5) / (3x + 2)
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Replace y with f-1(x):
f-1(x) = (4x - 5) / (3x + 2)
Therefore, the inverse of the rational function f(x) = (2x + 5) / (4 - 3x) is f-1(x) = (4x - 5) / (3x + 2).
