Let [.]denote the greatest integer function and f(x)=[x]+[2-x],-1≤x≤4.Then
f is not differentiable at x=3/2
f is differentiable at x=3
f is continues at x=0
f is not continues at x=1
f is continues at x=2
The Correct Option is
Solution and Explanation
Given that
The greatest integer function [x] represents the largest integer that is less than or equal to x.
For -1 ≤ x < 1: Since x is not an integer, both [x] and [2-x] will be 0. f(x) = [x] + [2-x] = 0 + 0 = 0
For 1 ≤ x < 2: In this interval, [x] will be 1, and [2-x] will be 0 (since 2-x is not an integer). f(x) = [x] + [2-x] = 1 + 0 = 1
For 2 ≤ x < 3: Both x and 2-x are integers in this interval. So, [x] = 2 and [2-x] = 0. f(x) = [x] + [2-x] = 2 + 0 = 2
For 3 ≤ x < 4: In this interval, [x] will be 3, and [2-x] will be 0 (since 2-x is not an integer). f(x) = [x] + [2-x] = 3 + 0 = 3
Now, let's consider the boundary points:
For \(x = -1: f(-1) = [-1] + [2 - (-1)] = -1 + 3 = 2\)
For \(x = 1: f(1) = [1] + [2 - 1] = 1 + 1 = 2\)
For \(x = 2: f(2) = [2] + [2 - 2] = 2 + 0 = 2\)\(\)
For \(x = 4: f(4) = [4] + [2 - 4] = 4 + 0 = 4\)
Finally, for \(x > 4\), both [x]\(\) and \([2 - x]\) will be equal to the integer part of \(x\) \(\), so \(f(x)\) will be the sum of these integer parts.
- For \(x > 4: f(x) = [x] + [2 - x] = x + (2 - x) = 2\)
So, the values of\(f(x)\) for \(-1 ≤ x ≤ 4\) can be represented as:\(f(x) = 2 \text{ for } x > 4\)
\(f(x) = 4 \text{ for } x = 4\)
\(f(x) = 3 \text{ for } 3 ≤ x < 4\)
\(f(x) = 2\) for \(2 ≤ x < 3\)
\(f(x) = 2\) for \(x = 2\)
\(f(x) = 1\) for \(1 ≤ x < 2\)
\(f(x) = 2\) for \(x = 1\)
\(f(x) = 2\) for \(x = -1\)
So, the correct option is (D) : f is continues at x=2.
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Concepts Used:
Relations and functions
A relation R from a non-empty set B is a subset of the cartesian product A × B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A × B.
A relation f from a set A to a set B is said to be a function if every element of set A has one and only one image in set B. In other words, no two distinct elements of B have the same pre-image.
Representation of Relation and Function
Relations and functions can be represented in different forms such as arrow representation, algebraic form, set-builder form, graphically, roster form, and tabular form. Define a function f: A = {1, 2, 3} → B = {1, 4, 9} such that f(1) = 1, f(2) = 4, f(3) = 9. Now, represent this function in different forms.
- Set-builder form - {(x, y): f(x) = y2, x ∈ A, y ∈ B}
- Roster form - {(1, 1), (2, 4), (3, 9)}
- Arrow Representation
