How To Find Domain Of Inverse Function

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How to Find the Domain of an Inverse Function

Introduction

An inverse function is a function that “undoes” another function. In other words, if you apply the inverse function to the output of the original function, you get the input back. For example, the inverse of the function f(x) = x + 1 is f-1(x) = x – 1, because f(f-1(x)) = x and f-1(f(x)) = x.

The domain of a function is the set of all possible input values. The range of a function is the set of all possible output values. When you find the inverse of a function, the domain and range of the original function become the range and domain of the inverse function, respectively.

Steps to Find the Domain of an Inverse Function

  1. Find the range of the original function.
  2. The domain of the inverse function is the range of the original function.

Example

Let’s find the domain of the inverse function of f(x) = x2.

Step 1: Find the range of the original function.
The range of f(x) = x2 is [0, ∞), because the square of any real number is always non-negative.

Step 2: The domain of the inverse function is the range of the original function.
Therefore, the domain of f-1(x) is [0, ∞).

Horizontal Line Test

There is another method to find the domain of an inverse function, called the horizontal line test.

Horizontal Line Test:
If a horizontal line intersects the graph of a function more than once, then the function is not one-to-one and does not have an inverse function.

For example, the function f(x) = x2 fails the horizontal line test because the horizontal line y = 1 intersects the graph at two points, (-1, 1) and (1, 1). Therefore, f(x) does not have an inverse function.

Conclusion

Finding the domain of an inverse function is a straightforward process. By following the steps outlined above, you can determine the set of all possible input values for the inverse function. The horizontal line test can also be used to verify whether a function has an inverse function.

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