Test of Mathematics at the 10+2 Level

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