Question

How do you verify that $ f\left( x \right) $ and $ g\left( x \right) $ are inverses given is, $ f\left( x \right) = x + 7 $ and $ g\left( x \right) = x - 7 $ .

Answer 1

Hint: The given question deals with the concept of functions. We can verify that $ f\left( x \right) $ and $ g\left( x \right) $ are inverse functions using two different methods- First, finding the inverse function of both the functions and Second, by finding composite function and then showing that the output of $ fog\left( x \right) = gof\left( x \right) $ equals to the input which is $ x $ . We will solve this question by finding the inverse function of both the given functions.

Complete step by step solution:
The given functions are $ f\left( x \right) = x + 7 $ and $ g\left( x \right) = x - 7 $ .
We will start solving this question by first finding out the inverse of the function $ f\left( x \right) = x + 7 $ .
From this function, we get the expression $ y = x + 7 $ . From this expression, we will try to calculate $ x $ .
 $
   \Rightarrow y = x + 7 \\
   \Rightarrow x = y - 7 \;
  $
We can clearly see that $ g\left( x \right) $ is the inverse of $ f\left( x \right) $ .
Now, let us look at the inverse of the function $ g\left( x \right) = x - 7 $ .
From this function, we get the expression $ y = x - 7 $ . From this expression, we will try to calculate $ x $ .
 $
   \Rightarrow y = x - 7 \\
   \Rightarrow x = y + 7 \;
  $
It is clearly visible that $ f\left( x \right) $ is the inverse of $ g\left( x \right) $ .
Therefore, if $ f $ is the inverse of $ g $ and $ g $ is the inverse of $ f $ , then $ f $ and $ g $ are inverse functions.
Hence, verified.

Note: The given question can also be solved by finding out the composite functions and then showing that the output of $ fog\left( x \right) = gof\left( x \right) $ equals to the input which is $ x $ .
The composite function $ fog\left( x \right) $ equals to-
 $
   \Rightarrow fog\left( x \right) = f\left( {g\left( x \right)} \right) \\
   \Rightarrow f\left( {g\left( x \right)} \right) = g\left( x \right) + 7 \\
   \Rightarrow f\left( {g\left( x \right)} \right) = x - 7 + 7 \\
   \Rightarrow f\left( {g\left( x \right)} \right) = x \;
  $
Similarly, the composite function $ gof\left( x \right) $ equals to-
 $
   \Rightarrow gof\left( x \right) = g\left( {f\left( x \right)} \right) \\
   \Rightarrow g\left( {f\left( x \right)} \right) = f\left( x \right) - 7 \\
   \Rightarrow g\left( {f\left( x \right)} \right) = x + 7 - 7 \\
   \Rightarrow g\left( {f\left( x \right)} \right) = x \;
  $
As we can see that $ fog\left( x \right) = gof\left( x \right) = x $ , therefore, we can conclude by saying that $ f\left( x \right) $ and $ g\left( x \right) $ are the inverse functions.
Hence, verified.