Question

$ f(x) $ and $ g(x) $ are linear functions such that for all $ x $ , $ f(g(x)) $ and $ g(f(x)) $ are identity functions. If $ f(0) = 4 $ and $ g(5) = 17 $ , compute $ f(2006) $

Answer 1

Hint: Linear function is a function with degree one. Write $ f(x) $ as a linear function. $ f(x) $ is called an identity function if $ f(x) = x $ $ \forall x \in \mathbb{R} $ . Write $ f(g(x)) $ as an identity function. Then put the values of $ x $ in the above formed functions according to the information given in the question. You will get your answer.

Complete step-by-step answer:
It is given in the question that, $ f(x) $ and $ g(x) $ are linear functions.
Thus we can write,
 $ f(x) = ax + b $ . . . (1)
and $ g(x) = px + q $ . . . (2)
A function $ f(x) $ is called identity function if, $ f(x) = x $ $ \forall x \in \mathbb{R} $
It is also given that, $ f(g(x)) $ and $ g(f(x)) $ are identity functions.
Thus, we can write
 $ f(g(x)) = g(x) $ . . . (3)
and $ g(f(x)) = f(x) $ . . . (4)
Now, it is given in the question that, $ f(0) = 4 $ and $ g(5) = 17 $ .
By substituting $ x = 0 $ in equation (1), we get
 $ f(0) = a(0) + b $
By substituting the value of $ f(0) = 4 $ , we get
 $ 4 = b $
Now, by substituting $ x = 5 $ in equation (3), we get
 $ f(g(5)) = g(5) $
 $ \Rightarrow f(17) = 17 $ $ \because \left( {g(5) = 17} \right) $
Now, again, by substituting $ x = 17 $ in equation (1), we get
 $ f(17) = a(17) + b $
By substituting the values of $ f(17) $ and b in the above equation, we get
 $ 17 = 17a + 4 $
Rearranging it we can write
 $ 17a = 13 $
 $ \Rightarrow a = \dfrac{{13}}{{17}} $
 $ \therefore f(x) = \dfrac{{13}}{{17}}x + 4 $
Now, by substituting $ x = 2006 $ in the above equation, we can write
 $ f(2006) = \dfrac{{13}}{{17}} \times 2006 + 4 $
 $ \Rightarrow f(2006) = 13 \times 118 + 4 $
 $ \Rightarrow f(2006) = 1538 $
Therefore, we can conclude that the value of $ f(2006) = 1538 $

Note: In this question, reading the question carefully and understanding it is important. You need to know the mathematical meaning of the words used in the question. This question is not so difficult if you know what is linear function and what is identity function. Writing $ f(x) $ in terms of a linear function helped in getting the general form of $ f(x) $ . In which we could substitute $ x = 2006 $ to get our answer.