If \(f:R\to R\) is defined by \(f(x)=x^2-3x+2,find \,f(f(x)).\)
It is given that \(f:R\to R \,is\,defined \,as f(x)=x^2-3x+2\).
\(f(f(x))=f(x^2-3x+2)\)
= \((x^2-3x+2)^2-3(x^2-3x+2)+2\)
= \(x^4+9x^2+4-6x^3-12x+4x^2-3x^2+9x-6+2\)
= \(x^4-6x^3+10x^2-3x\).
A school is organizing a debate competition with participants as speakers and judges. $ S = \{S_1, S_2, S_3, S_4\} $ where $ S = \{S_1, S_2, S_3, S_4\} $ represents the set of speakers. The judges are represented by the set: $ J = \{J_1, J_2, J_3\} $ where $ J = \{J_1, J_2, J_3\} $ represents the set of judges. Each speaker can be assigned only one judge. Let $ R $ be a relation from set $ S $ to $ J $ defined as: $ R = \{(x, y) : \text{speaker } x \text{ is judged by judge } y, x \in S, y \in J\} $.
During the festival season, a mela was organized by the Resident Welfare Association at a park near the society. The main attraction of the mela was a huge swing, which traced the path of a parabola given by the equation:\[ x^2 = y \quad \text{or} \quad f(x) = x^2 \]