Functional Equation (Tomato subjective 64)

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

One Reply to “Functional Equation (Tomato subjective 64)”

  1. 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

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