What would be the inverse function for the following condition?

Question

What would be the inverse function condition for the above question.

Answer 1

Hint: Replace $x$ by $1/x$ to find an expression for $f(x)$. Then find the inverse.

Answer 2

You can rearrange the equation to give: $$2(f(x)-2x)=-\left(f\left(\frac 1x\right)-\frac 2x\right)$$

Now let $g(x)=f(x)-2x$ and this becomes $$g(x)=-\frac 12g\left(\frac 1x\right)$$And with $y=\frac 1x$ $$g\left(\frac 1y\right)=-\frac 12g\left(y\right)$$

You just need to put the pieces together. Really though you might start by playing with $x=2, \frac 12, 3, \frac 13$ and see if you can spot a pattern that you can prove.

Answer 3

2 y(x) + y(1/x) = 4 x + 2/x

First and foremost you need to solve for y(x) and there is no way out of it...

and that is the important implicated part of this question !

By inspection/comparison/equating of terms with argument and its reciprocal, y = 2 x.

To get inverse function swap x and y.

$ x = 2 y, y= x/2 ; i.e., f^{-1}(x) = x/2; $

At x=4 the inverse function evaluates to 4/2 = 2.