Relations and Functions

Maths · Class 11

Simulation · Maths · Class 11

Four standard graphs

"Four standard graphs" is a free, interactive Class 11 Maths simulation for JEE, built on the lesson "Signum and greatest integer functions" in Relations and Functions. Choose |x|, sgn x, [x] or 1/x, then slide x. The orange dot sits on the graph at (x, f(x)); a filled dot on the graph is included and a hollow dot is not. |x| has range [0, ∞); sgn x has range {−1, 0, 1} with sgn 0 = 0; [x] has range Z and jumps at every integer; 1/x has domain and range R − {0}.

From the lesson Signum and greatest integer functions in Relations and Functions. Change the values and watch what happens.

Four standard graphsMaths · Class 11

The idea behind it

NCERT §2.4.1

  • The signum function f : R → R gives the sign of x: f(x) = 1 if x > 0, f(0) = 0, and f(x) = −1 if x < 0.
  • The domain of the signum function is R and its range is {−1, 0, 1}. Its graph is two horizontal rays at heights 1 and −1, with open ends at x = 0, and the single point (0, 0).
  • For x ≠ 0 the signum function equals |x|/x, which is 1 for positive x and −1 for negative x; the formula |x|/x itself is undefined at 0.
  • The greatest integer function f : R → R, f(x) = [x], gives the greatest integer less than or equal to x. [3.7] = 3, [5] = 5 and [−2.4] = −3, not −2.
  • For every integer n, [x] = n exactly when n ≤ x < n + 1. So [x] = 0 on [0, 1), [x] = 1 on [1, 2) and [x] = −1 on [−1, 0). Its domain is R and its range is Z.
  • The graph of [x] is a staircase of horizontal steps of length 1. Each step includes its left end (filled dot) and excludes its right end (hollow dot), where the value jumps up by 1.
  • For every real x, [x] ≤ x < [x] + 1, and [x] = x exactly when x is an integer.

Questions

2 answers

What does the "Four standard graphs" simulation show?
Choose |x|, sgn x, [x] or 1/x, then slide x. The orange dot sits on the graph at (x, f(x)); a filled dot on the graph is included and a hollow dot is not.
Is this simulation free?
Yes. It is free and runs in the browser on a phone or a laptop. It belongs to the lesson "Signum and greatest integer functions" in Relations and Functions (Class 11 Maths).

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