Which element of domain has the image 1 in the function \( f(x)=\frac{2 \tan x}{1+\tan x} \) ?

To determine which element of the domain results in an image of \(1\) in the function \( f(x)=\frac{2 \tan x}{1+\tan x} \), we need to solve the equation \( f(x) = 1 \). This leads to the equation \( \frac{2 \tan x}{1 + \tan x} = 1 \). Simplifying gives: \[ 2 \tan x = 1 + \tan x \] Subtracting \( \tan x \) from both sides results in: \[ \tan x = 1 \] The solutions for \( \tan x = 1 \) occur when \( x = \frac{\pi}{4} + n\pi \) for \( n \in \mathbb{Z} \), meaning that any input of the form \( \frac{\pi}{4} + n\pi \) will produce an output of \(1\) for the function \( f(x) \). Now, let's have some fun with the math! Picture this: you're at a math party, and the host asks who can find the angle where the tangent equals one; suddenly, someone shouts, "It’s party time at \(\frac{\pi}{4}\)!" But wait, there's more - anyone who shows up exactly \(n\) whole rotations later (like ever-trendy partygoers) is also welcome to this exclusive club of angles that yield \(1\). Such is the quirky nature of trigonometric functions!