First slide

Functions (XII)

Question

$The number of linear functions f satisfying f (x + f (x))$ $= x + f (x) \forall x \in R is$

Moderate

A

0
B

1
C

2
D

3

Solution

$Let f (x) = a x + b - - - - - - (1)$
$\begin{aligned} \Rightarrow f (a x + b + x) = x + a x + b \Rightarrow f ((a + 1) x + b) = (a + 1) x + b \\ Replace (a + 1) x + b by y, we have \Rightarrow f (y) = (a + 1) (\frac{y - b}{a + 1}) + b \end{aligned}$
$or f (x) = (a + 1) (\frac{x - b}{a + 1}) + b - - - - - (2)$
$Now compare (1) and (2), we get a = \pm 1, b = 0$
There are two linear functions. Hence, the least value is 2.

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