# 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.

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.