Math Lesson 16.6.1 - Definition of Inverse Function

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Welcome to our Math lesson on Definition of Inverse Function, this is the first 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.

Definition of Inverse Function

In all functions discussed so far, we have expressed the dependent variable y [in functions terminology we call it f(x)] in terms of the independent variable x. Let's suppose you are given the task to express the variable x in terms of f(x). What would you do in this case?

For example, the task is to express the variable x in the function f(x) = 1 - 3x in terms of f(x) (or y). In other words, we must isolate the variable x and express it in terms of the rest of the expression. The procedure is as follows:

If we switch the places of x and f(x) in the above equation, we obtain the inverse of the function f(x), which we denote as f - 1(x). Thus, we have

By definition, an inverse function - otherwise known as an anti-function - is defined as a function, which can reverse into another function.

In other words, an inverse function is a function that undoes the action of another function.

From the above definition, it is clear that the inverse of the inverse gives the original function, i.e.

Indeed, expressing f - 1[f - 1(x)] = y in the above example, we have

Switching the places of x and y in the last equation yields

or

Example 1

Two functions, f(x) = 2x - 1 and g(x) = 1 - 3x are given. Calculate:

1. f - 1 (3)
2. g - 1(2)
3. g - 1 ◦ f - 1 (4)

Solution 1

1. We have
   f(x) = 2x - 1
   2x = f(x) + 1
   x = f(x) + 1/2
   f^-1 (x) = x/2 + 1/2
   Thus,
   f^-1 (3) = 3/2 + 1/2
   f^-1 (3) = 4/2
   f^-1 (3) = 2
2. We use the same approach as above to find g - 1(2). We have
   g(x) = 1 - 3x
   3x = 1 - g(x)
   x = 1 - g(x)/3
   g^-1 (x) = 1 - x/3
   g^-1 (x) = -x/3 + 1/3
   Thus,
   g^-1 (2) = -2/3 + 1/3
   g^-1 (2) = -1/3
3. First, we find g - 1 ◦ f - 1 (x). We have
   g^-1 ∘ f^-1 (x) = g^-1 [f^-1 (x)]
   = -x/2 + 1/2/3 + 1/3
   = x + 1/2/3 + 1/3
   = (x + 1)/2 ⋅ 1/3 + 1/3
   = x + 1/6 + 2/6
   = x + 3/6
   Thus,
   g^-1 ∘ f^-1 (4) = 4 + 3/6
   = 7/6

You have reached the end of Math lesson 16.6.1 Definition of Inverse 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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