Finding the inverse of a function can be challenging, especially when the function includes fractions. In this article, we’ll go through the step-by-step process of finding the inverse of a function with a fraction. By the end of this article, you’ll have a better understanding of how to approach inverse functions with fractions and be equipped with the skills to solve similar problems.
What is an Inverse Function?
An inverse function is simply the reverse of a function. It is a function that “undoes” the original function. For example, if f(x) = 3x – 1, then the inverse of f(x) would be f^-1(x) = (x+1)/3. The inverse function takes the output of the original function as input and returns the original input as output.
Step-by-Step Process
To find the inverse of a function with a fraction, we can follow these steps:
Step 1: Replace f(x) with y
The first step is to replace f(x) with y. This is a common step in finding inverse functions, and it makes the process easier to follow.
Step 2: Swap x and y
The second step is to swap x and y. This is where the inverse function comes in. Instead of plugging in x and getting y, we’ll plug in y and get x.
Step 3: Solve for y
The third step is to solve for y. This is where the fraction comes in. You’ll need to be comfortable manipulating fractions to solve for y.
Let’s walk through an example to see how these steps work in practice.
Example
Suppose we have the function f(x) = (2x – 1)/(x + 3). Our goal is to find the inverse of this function.
Step 1: Replace f(x) with y
y = (2x – 1)/(x + 3)
Step 2: Swap x and y
x = (2y – 1)/(y + 3)
Step 3: Solve for y
We’ll start by multiplying both sides of the equation by (y + 3) to get rid of the fraction in the denominator:
x(y + 3) = 2y – 1
Expanding the left-hand side gives us:
xy + 3x = 2y – 1
Moving all the y terms to the left-hand side and all the x terms to the right-hand side gives us:
2y – xy = 3x – 1
Factoring out y on the left-hand side gives us:
y(2 – x) = 3x – 1
Dividing both sides by (2 – x) gives us:
y = (3x – 1)/(2 – x)
This is our inverse function!
Conclusion
Finding the inverse of a function with a fraction can be challenging, but by following these steps, it becomes much more manageable. Remember to replace f(x) with y, swap x and y, and solve for y. With practice, you’ll become more comfortable with manipulating fractions and solving for y.
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