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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## Problem

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).

## Solution:

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

In the original equation if we put x=-x then we will have two equations in f(x) and f(-x). and then we can solve the two equations and find f(x)=4x+3