How Cheenta works to ensure student success?
Explore the Back-Story

Graphing integer value function | Tomato Subjective 117

This is a subjective problem from TOMATO based on Graphing integer value function.

Problem: Graphing integer value function

Let [x] denote the largest integer (positive, negative or zero) less than or equal to x. Let y= f(x) = [x] + \sqrt{x - [x]} ,s=2 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 ,s=2
(ii) Show that, any given real number y_0 ,s=2, there is a real number x_0 ,s=2 such that y_0 = f(x_0) ,s=2

Discussion:

First note that \sqrt{x - [x]} ,s=2 is same as \sqrt{t} , 0\le t \le 1 ,s=2.

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} ,s=2 , t\in(0,1) ,s=2. 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 ,s=2. We take x_0 = [y_0] + (y - [y_0])^2 ,s=2. Then f(x_0) = [x_0] + \sqrt(x - [x_0] = [y_0] + \sqrt{(y - [y_0])^2} = y_0 ,s=2 (since 0 \le (y - [y_0]) < 1 \Rightarrow 0 \le (y - [y_0])^2 < 1 ,s=2 )

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

This is a subjective problem from TOMATO based on Graphing integer value function.

Problem: Graphing integer value function

Let [x] denote the largest integer (positive, negative or zero) less than or equal to x. Let y= f(x) = [x] + \sqrt{x - [x]} ,s=2 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 ,s=2
(ii) Show that, any given real number y_0 ,s=2, there is a real number x_0 ,s=2 such that y_0 = f(x_0) ,s=2

Discussion:

First note that \sqrt{x - [x]} ,s=2 is same as \sqrt{t} , 0\le t \le 1 ,s=2.

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} ,s=2 , t\in(0,1) ,s=2. 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 ,s=2. We take x_0 = [y_0] + (y - [y_0])^2 ,s=2. Then f(x_0) = [x_0] + \sqrt(x - [x_0] = [y_0] + \sqrt{(y - [y_0])^2} = y_0 ,s=2 (since 0 \le (y - [y_0]) < 1 \Rightarrow 0 \le (y - [y_0])^2 < 1 ,s=2 )

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

Leave a Reply

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Knowledge Partner

Cheenta is a knowledge partner of Aditya Birla Education Academy
Cheenta

Cheenta Academy

Aditya Birla Education Academy

Aditya Birla Education Academy

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.
JOIN TRIAL
support@cheenta.com
Menu
Trial
Whatsapp
magic-wandrockethighlight