Math Lesson 16.6.3 - Methods used for Obtaining the Inverse of a Function

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Welcome to our Math lesson on Methods used for Obtaining the Inverse of a Function, this is the third lesson of our suite of math lessons covering the topic of Inverse Functions, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Methods used for Obtaining the Inverse of a Function

There are two analytical methods used to find the inverse of a function but they share the same procedure and obviously, bring the same output. We covered the first method in the previous example, i.e. expressing f(x) (or y) in terms of x and swapping the places of x and f(x) at the end of the procedure.

The second method consists of swapping the position of x and f(x) immediately; then we continue with the rest of the operations. For convenience, we can express f(x) by y, as they mean the same thing. Let's explain this method through an example.

Example 3

Find the inverse of the following functions.

1. f(x) = 3x + 3
2. g(x) = 2x - 3/5

Solution 3

1. We can immediately swap the positions of x and f(x), and then we continue with the rest of the solution. Thus,
   f(x) = 3x - 3
   y = 3x - 3
   x = 3y - 3
   x + 3 = 3y
   y = x + 3/3
   Thus, the inverse function of f(x) is
   f^-1 (x) = x + 3/3
   We can proof this result using the other method, i.e. expressing f(x) in terms of x and swapping the places of x and f(x) at the end of the process. We have
   f(x) = 3x - 3
   f(x) + 3 = 3x
   x = f(x) + 3/3
   f^-1 (x) = x + 3/3
2. Using the same method as in (a) yields
   g(x) = 2x - 3/5
   y = 2x - 3/5
   x = 2y - 3/5
   5x = 2y - 3
   2y = 5x + 3
   y = 5x + 3/2
   Thus, the inverse function g - 1(x) is
   g^-1 (x) = 5x + 3/2
   Proof:
   g(x) = 2x - 3/5
   5g(x) = 2x - 3
   5g(x) + 3 = 2x
   x = 5g(x) + 3/2
   Thus, the inverse function of g(x) is
   g^-1 (x) = 5x + 3/2
   Again, we obtained the same result with both methods.

You have reached the end of Math lesson 16.6.3 Methods used for Obtaining the Inverse of a Function. There are 6 lessons in this physics tutorial covering Inverse Functions, you can access all the lessons from this tutorial below.

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