2 Inverse of a function

We have seen that a function can be regarded as taking an input, x , and processing it in some way to produce a single output f ( x ) as shown in Figure 16(a).

Figure 16 :

{ The second block reverse the process in the first}

A natural question to ask is whether we can find another function that will reverse the process. In other words, can we find a function that will start with f ( x ) and process it to produce x again? This idea is also shown in Figure 16(b). If we can find such a function it is called the inverse function to f ( x ) and is given the symbol f − 1 ( x ) . Do not confuse the ‘ − 1 ’ with an index, or power. Here the superscript is used purely as the notation for the inverse function. Note that the composite function f − 1 ( f ( x ) ) = x as shown in Figure 17.

Figure 17 :

{ $f^{-1}$ reverses the process in $f$}

Example 6

Find the inverse function to f ( x ) = 3 x − 8 .

Solution

The given function takes an input, x and produces an output 3 x − 8 . The inverse function, f − 1 , must take an input 3 x − 8 and give an output x . That is

f − 1 ( 3 x − 8 ) = x

If we introduce a new variable z = 3 x − 8 , and transpose this for x to give

x = z + 8 3 then f − 1 ( z ) = z + 8 3

So the rule for f − 1 is add 8 to the input and divide the result by 3. Writing f − 1 with x as its argument gives

f − 1 ( x ) = x + 8 3

This is the inverse function.

Not all functions possess an inverse function. In fact, only one-to-one functions do so. If a function is many-to-one the process to reverse it would require many outputs from one input contradicting the definition of a function.

Task!

Find the inverse of the function f ( x ) = 7 − 3 x , using the fact that the inverse function must take an input 7 − 3 x and produce an output x . So f − 1 ( 7 − 3 x ) = x

Introduce a new variable z so that z = 7 − 3 x and transpose this to find x . Hence write down the inverse function:

f − 1 ( z ) = 7 − z 3 . With x as its argument the inverse function is f − 1 ( x ) = 7 − x 3 .

Exercises
  1. Explain why a one-to-many rule cannot be a function.
  2. Illustrate why y = x 4 is a many-to-one function by providing a suitable example.
  3. By sketching a graph of y = 3 x − 1 show that this is a one-to-one function.
  4. Explain why a many-to-one function does not have an inverse function. Give an example.
  5. Find the inverse of each of the following functions:
    1. f ( x ) = 4 x + 7 ,
    2. f ( x ) = x ,
    3. f ( x ) = − 23 x ,
    4. f ( x ) = 1 x + 1 .
    1. f − 1 ( x ) = x − 7 4 ,
    2. f − 1 ( x ) = x ,
    3. f − 1 ( x ) = − x 23 ,
    4. f − 1 ( x ) = 1 − x x .