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THE GREATEST INTEGER FUNCTION

THE GREATEST INTEGER FUNCTION. f ( x ) = x = n if n ≤ x < n+ 1 , n is an integer. Domain of f ( x ) is the set of all real numbers ( − ∞ , ∞ ) Range of f ( x ) is the set of integers { …, -3, -2, -1, 0, 1, 2, 3, …}

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THE GREATEST INTEGER FUNCTION

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  1. THE GREATEST INTEGER FUNCTION f(x) = x = n if n ≤ x < n+1, n is an integer Domain of f (x) is the set of all real numbers (− ∞, ∞) Range of f (x) is the set of integers { …, -3, -2, -1, 0, 1, 2, 3, …} int(x) is also used for the greatest integer function.

  2. THE GREATEST INTEGER FUNCTION f(x) = x = n if n ≤ x < n+1n is an integer Examples. n = n for any integer n 2 < 5 < 3 5 = 2because -3 <- 5 < -2 -5 = -3because e = 2, 0.99 = 0, π = 3, 2.98 = 2, -2.98 = -3 3.99 = 3, 3.01 = 3, -e = -3 ,-π = -4, -3 = -3

  3. TRUE or FALSE • x ≤ x for all x . • x < x +1 for all x . • x = kx for all x and k . • x+y = x + y for all x and y . • ― = for all x and y . • for all x and y . x x y y

  4. THE GREATEST INTEGER FUNCTION y f(x) = x 4 3 2 1 x 0 -4 -1 -2 1 2 4 -3 3 -2 -3 -4

  5. Exercise 1 If f (x) = int(x), then what is the domain of ?

  6. Exercise 2 Solve the equation int(2x-3) = -5

  7. Exercise 3 Solve the equation 3int(4 – 5x) +5 = 0

  8. Exercise 4 Sketch the graph of f (x) = int(x)/|x| for -4 ≤ x ≤ 4

  9. f (x) = int(x)/|x| 1 -4 4 -3 -2 -1 1 2 3 -1 -2

  10. Exercise 5 Find the range of f ( x ) = int(2x-1) if 2 ≤ x ≤ 6

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