# Graphing integer value function (Tomato Subjective 117)

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 I.S.I. & C.M.I. course, discusses these topics in the ‘Calculus’ module.
• Book Suggestions: Play with Graphs, Arihant Publication