CAT (QA) PYQs

Let $f(x) = x^2$ and $g(x) = 2^x$, for all real $x$. Then the value of $f(f(g(x)) + g(f(x)))$ at $x = 1$ is

Let r be a real number and $f(x) = \begin{cases} 2x-r & \text{if } x \ge r \\ 1 & \text{if } x < r \end{cases}$. Then, the equation $f(x) = f(f(x))$ holds for all real values of x where.

For any non-zero real number x, let $f(x) + 2 f(\frac{1}{x}) = 3x$. Then, the sum of all possible values of $x$ for which $f(x) = 3$, is