* 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).

*: 2f(x) + 3f(-x) = 15 – 4x*

**Solution**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