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Functions (XII)

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Question

$f : R \to R f (x^{2} + x + 3) + 2 f (x^{2} - 3 x + 5) = 6 x^{2} - 10 x + 17 \forall x \in R then the value of f (100) is$

Moderate

Solution

$obviously f is a linear polynomial (Since degree of LHS and RHS is same)$
$\begin{array}{l} let f (x) = a x + b hence f (x^{2} + x + 3) + 2 f (x^{2} - 3 x + 5) \\ \equiv 6 x^{2} - 10 x + 17 \\ \Rightarrow [a (x^{2} + x + 3) + b] + 2 [a (x^{2} - 3 x + 5) + b] \\ \equiv 6 x^{2} - 10 x + 17 \\ c o m p a r i n g c o e f f o f x^{2} and coeff. of x \\ \Rightarrow a + 2 a = 6 . . . . . . (1) \\ \Rightarrow a - 6 a = - 10 . . . . . . (2) a = 2 \\ c o m p a r i n g c o n s \tan t s \\ 3 a + b + 10 a + 2 b = 17 \\ \Rightarrow 6 + b + 20 + 2 b = 17 \\ \Rightarrow b = - 3 \\ ∴ f (x) = 2 x - 3 \\ \Rightarrow f (100) = 197 \end{array}$

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