**Problem: **Let [x] denote the largest integer (positive, negative or zero) less than or equal to x. Let \(y= f(x) = [x] + \sqrt{x – [x]} \) be defined for all real numbers x.

(i) Sketch on plain paper, the graph of the function f(x) in the range \(-5 \le x \le 5 \)

(ii) Show that, any given real number \(y_0 \), there is a real number \(x_0 \) such that \(y_0 = f(x_0) \)

**Discussion: **

First note that \(\sqrt{x – [x]} \) is same as \(\sqrt{t} , 0\le t \le 1 \).

It’s graph between 0 to 1 looks like:

Clearly [x] part only increments (or decrements) it by integer quantity as [x] is constant between any two integers. That for any integer k for all \( x \in (k, k+1) \).

\(f(x) = k +\sqrt{t} \) , \(t\in(0,1) \). Hence graph of f(x) is as follows:

Finally consider and arbitrary value \(y_0 \). We take \(x_0 = [y_0] + (y – [y_0])^2 \). Then \(f(x_0) = [x_0] + \sqrt(x – [x_0] = [y_0] + \sqrt{(y – [y_0])^2} = y_0 \) (since \(0 \le (y – [y_0]) < 1 \Rightarrow 0 \le (y – [y_0])^2 < 1 \) )

## Chatuspathi:

**What is this topic:**Graphing of functions**What are some of the associated concept:**Greatest Integer Function**Where can learn these topics:**Cheenta**Book Suggestions:**Play with Graphs, Arihant Publication