CAT (QA) PYQs

Let $f(x) = \dfrac{x}{(2x-1)}$ and $g(x) = \dfrac{x}{(x-1)}$. Then, the domain of the function $h(x) = f(g(x)) + g(f(x))$ is all real numbers except

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.

The number of the real roots of the equation $2\cos (x(x + 1)) = 2^x + 2^{-x}$ is