Introduction to The Inverse of a Function
Let f be the function which assigns to each working adult American his or her Social Security Number (a 9digit string of integers). Let g be the function which assigns to each working adult American his or her age in years (an integer between 0 and 150). Both functions have the same domain, and both take values in the nonnegative integers. But there is a fundamental difference between f and g . If you are given a Social Security number, then you can determine the person to whom it belongs. There will be one and only one person with that number. But if you are given a number between 0 and 150, then there will probably be millions of people with that age. You cannot identify a person by his/her age. In summary, if you know g (x) then you generally cannot determine what x is. But if you know f(x) then you can determine what (or who) x is. This leads to the main idea of this subsection.
Let f : S → T be a function. We say that f has an inverse (is invertible) if there is a function f ^{−1} : T → S such that (f ο f ^{−1})(t) = t for all t T and ( f ^{−1} ο f )(s) = s for all s S . Notice that the symbol f ^{−1} denotes a new function which we call the inverse of f.
Basic Rule for Finding Inverses:
To find the inverse of a function f, we solve the equation
(f ο f ^{−1})(t) = t
for the function f ^{−1} ( t ).
Examples
Example 1
Find the inverse of the function f(s) = 3 s .
Solution 1
We solve the equation
(f ο f ^{−1})(t) = t.
This is the same as
f (f ^{−1}(t)) = t .
We can rewrite the last line as
Thus f ^{−1}(t) = t /3.
Example 2
Let be defined by f (s) = 3 s^{5}. Find f ^{−1}
Solution 2
We solve
You Try It: Find the inverse of the function
It is important to understand that some functions do not have inverses.
Example 3
Let be defined by f (s) = s^{2}. If possible, find f ^{−1}.
Solution 3
Using the Basic Rule, we attempt to solve
(f ο f ^{−1})(t) = t .
Writing this out, we have
[f ^{−1} (t)] ^{2} = t .
But now there is a problem: we cannot solve this equation uniquely for f ^{−1}( t ). We do not know whether . Thus f ^{−1} is not a well defined function. Therefore f is not invertible and f ^{−1} does not exist.
Math Note: There is a simple device which often enables us to obtain an inverse—even in situations like Example 1.42. We change the domain of the function. This idea is illustrated in the next example.
Example 4
Define by the formula . Find .
Solution 4
We attempt to solve
Writing this out, we have
This looks like the same situation we had in Example 1.42. But in fact things have improved. Now we know that must , because must have range S = { s : s ≥ 0}. Thus is given by .
You Try It: The equation y = x^{2} + 3x does not describe the graph of an invertible function. Find a way to restrict the domain so that it is invertible.
Now we consider the graph of the inverse function. Suppose that f : S → T is invertible and that (s , t ) is a point on the graph of f . Then t = f (s) hence s = f ^{−1}(t) so that ( t , s ) is on the graph of f ^{−1}. The geometrical connection between the points (s , t) and (t , s) is exhibited in Fig. 1.47: they are reflections of each other in the line y = x. We have discovered the following important principle:
The graph of f ^{−1} is the reflection in the line y = x of the graph of f. Refer to Fig. 1.48.

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