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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:

Screen Shot 2015-11-29 at 9.40.45 PM

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:

Screen Shot 2015-11-29 at 9.47.31 PM

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
November 30, 2015

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