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# Inverse of a Function

### By juan taun

The inverse of a function is a function that, when given an output of the original function, returns the input. The domain and range of the original function are the range and domain of the function inverse, respectively. The domain of a function is the set of all numbers which can be inputted into the function, and the range of a function is the set of all possible outputs of the function. The function inverse is usually notated f^{-1}.

## Finding the Inverse

To find the inverse of a function, substitute y for f(x), and solve for x. Then, replace the the isolated x with f^{-1}(x) and replace the y with x. For example, let's find the inverse of f(x)=3x+4. First, we substitute y for f(x), resulting in y=3x+4. Second, we solve for x. We start by subtracting 4 from each site, yielding y-4=3x. Next, we divide both sides by 3, which gets us 1/3y-4/3=x. Third, we replace x with f^{-1}(x) and replace the y's with x's, resulting in f^{-1}(x)=1/3x-4/3.

We can test this inverse by plugging in a number for x. Let's say x is 4.

Therefore, f(4)=3(4)+4, and f(4)=16

Now, we can plug 16 into the inverse function. f^{-1}(16)=1/3(16)-4/3

Therefore, f^{-1}(16)=16/3-4/3, and f^{-1}(16)=12/3=4.

Since placing the result of the function into the inverse function results in the input, we correctly derived the inverse!