Understanding the Concept of Inverse Functions

Let us consider the functions

f(x) = x^3  and g(x) = \sqrt[3]{x}.

The table on the left shows the ordered pairs (x,f(x)). We used f(x) and substitute them to x in the second table.

inverse functions

As we can see, all the values of g(x) are the same as those of x in the first table. For example,  if we have x = 3 in the first table, applying f, we get  27 as output. On the other hand, if we apply g to 27, the value returns to 3

3 ——— f ———>27 ———-g ———>3

From the tables above, we can see that it also does this to other values. Thus, we can say that it seems that the function g is an”undo” for  f.

In mathematics, functions such as f and g, are called inverse functions.  In mathematical notation, if f(x) = x^3 and g(x) = \sqrt[3]{x} are inverse functions, we can replace the notation of g as f^{-1} making it f^{-1}(x) = \sqrt[3]{x} From this notation we can say that f^{-1} is the inverse of f.

Please take note that f^{-1} is not the same as f raised to negative one.

Let us try to look at the ordered pairs above from the two functions.

(-3,-27) and  (-27,-3)

(-2, -8) and (-8, -2)

(-1, -1) and (-1, -1)

and so on, we can say that the if f has ordered pairs (a,b) its corresponding ordered pair in f^{-1} is (b,a).

From the discussion above and looking at the tables, we can see that

(1) the domain of f^{-1} is the same as the range of f.

(2) the range of f^{-1} is the same as the domain of f.

(3) the corresponding pair of (a,b) in f is (b,a) in f^{-1}.

From the discussion above, we can ask the following questions.

(1) What are the properties of inverse functions and its graphs?

(2) Do all functions have inverses?

(3) If a function has an inverse, how do we find it?

These are the topics that we are discuss in the next few posts.

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