The function $f$ has property: $f(x)+f(1/x)=x$ and what is the largest set of real numbers?

Question

Problem. $f(x)+f\left(\frac{1}{x}\right)=x$ and we need to figure out the largest set of real numbers that can be the domain of $f$.

My steps:

Step 1. I substituted $x$ with $1/x$ to replace the reciprocal signs as shown below: $f(1/x)+f(x)=\frac{1}{x}$ that's where I got stuck.

Answer 1

$$f(x)+f\left(\dfrac{1}{x}\right)=x\tag{1}$$

Replacing $x$ with $\dfrac{1}{x}$, we get $$f\left(\dfrac{1}{x}\right)+f(x)=\dfrac{1}{x}\tag{2}$$

From $(1)$ and $(2)$, we get

$$x=\dfrac{1}{x} \Rightarrow x=\pm 1$$