Question

If a function satisfies a relation 2f(x) + f(1/x) = -x then f(2) equals.....​

Answer 1

Given,

A function satisfying the relation 2f(x) + f(1/x) = -x

To Find,

The value of f(2)

Solution,

Putting x = 2 in given function

2f(2) + f(1/2) = -2

f(1/2) = -2-2f(2)

Now, putting =1/2

2f(1/2)+f(2)=-1/2

-2-2f(2)+f(2) = -1/2

-3/2 = f(2)

Hence, the value of f(2) is equal to -3/2.

Answer 2

Given: $2f(x)+f(\frac{1}{x})=-x$

Substitute $x$ equals  2,

$2f(2)+f(\frac{1}{2})=-2$  

$f(\frac{1}{2})=-2-2f(2)$........(1)

Substitute $x$ equals to $\frac{1}{2}$,

$2f(\frac{1}{2})+f(\frac{1}{\frac{1}{2}})=-2$

$2f(\frac{1}{2})+f(2)=-2$

$f(2)=-2-2f(\frac{1}{2})$ ........(2)

Substitute $f(\frac{1}{2})$ value from eq (1) to eq(2),

$f(2)=-2-2(-2-2f(2))\\f(2)=-2+4+4f(2)\\f(2)=2+4f(2)\\3f(2)=-2\\f(2)=\frac{-2}{3}$