Given that the function f (x) satisfies f (x) = 2F (1 / x) + X, find the expression of F (x)

Given that the function f (x) satisfies f (x) = 2F (1 / x) + X, find the expression of F (x)

solution
f(x)=2f(1/x)+x ①
Let x = 1 / X
Then f (1 / x) = 2F (x) + 1 / X (2)
② 2
2f(1/x)=4f(x)+2/x ③
① + 3
f(x)+2f(1/x)=2f(1/x)+4f(x)+x+(2/x)
∴3f(x)+x+2/x=0
∴f(x)=-x/3-2/3x

The difference between function f (x) e ^ LNX and function f (x) = ln e ^ x

Different domains
The definition field of F (x) = e ^ LNX is x > 0
The domain of G (x) = lne ^ x is x ∈ R

Why ln (1 / x) is equal to - LNX, the solving process

ln(1/x)=ln(x^(-1))
In logarithm, ln (x ^ a) = a · LNX
ln(1/x)=ln(x^(-1))=-1·lnx=-lnx
If it is a proof, then
Let ln (1 / x) = a, LNX = B,
Then e ^ a = 1 / x, e ^ B = X
x=1/(e^a)=e^(-a)=e^b
Then - a = b
That is - ln (1 / x) = LNX
Ln (1 / x) = - LNX