This is a Test of Mathematics Solution Subjective 64 (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance.

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If f(x) is a real-valued function of a real variable x, such that 2f(x) + 3f(-x) = 15 – 4x for all x, find the function f(x).

2f(x) + 3f(-x) = 15 – 4x Putting x =0, we gets 2f(0) + 3f(0) = 15 Or, f(0) = 3. Let g(x) = f (x) – 3 2f(x) + 3f(-x) = 15 – 4x Or, 2g(x) + 3g(-x) = – 4x …(i) Now put x = – x 2g(-x) + 3g(x) = 4x …(ii) (i) + (ii), g(x) + g(-x) = o Or, g(x) = -g(-x) …(iii)

Now 2g(x) + 3g(-x) = – 4x Or, –g(x) = – 4x [from (iii)] Or, g(x) = 4x Or, f(x) = 4x + 3