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Functions: Number Linear Functions Satisfying

JEE Maths question with a full step-by-step solution.

Question
The number of linear functions satisfying f[x+f(x)]=x+f(x)f[x+f(x)] = x+f(x) for all x∈Rx \in \mathbb{R} is.
Solution
Answer: 2
Step 1: Set up the equation for f(x)=mx+cf(x) = mx+c
f(x+f(x))=m(x+mx+c)+c=m(1+m)x+mc+cf(x+f(x)) = m(x+mx+c)+c = m(1+m)x+mc+c
x+f(x)=(1+m)x+cx+f(x) = (1+m)x+c
 Step 2: Match coefficients Coefficient of xx: m(1+m)=1+m⇒(m−1)(m+1)=0⇒m=±1m(1+m) = 1+m \Rightarrow (m-1)(m+1) = 0 \Rightarrow m = \pm 1. Constant term: mc+c=c⇒cm=0mc+c = c \Rightarrow cm = 0. For m=1m = 1: c=0c = 0, giving f(x)=xf(x) = x. For m=−1m = -1: c=0c = 0, giving f(x)=−xf(x) = -x. Both satisfy the original equation. Number of linear functions = 2. Answer: 2
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