Functions - Calculus - Mathematics Previous Year Questions

Let $[x]$ denote the greatest integer less than or equal to $x$. The number of positive integer solutions of the equation

$$\left[\frac{x}{19}\right]=\left[\frac{x}{20}\right]$$

are:

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ be two functions such that

$$g \circ f(x)= \begin{cases}x^2, & \text { if } x \geq 0 \\ e^x-1, & \text { if } x<0\end{cases}$$

Then which of the following statements is correct?

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be the function defined by

$$f(x)= \begin{cases}x^2-4 x-5 & \text { if } x \geq 1, \\ 2 x & \text { if } x<1 .\end{cases}$$

Which one of the following statements is TRUE?