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Let g be the greatest integer function. Then the function f(x) = (g(x))2− g(x2) is discontinuous at
- a)All integers
- b)All integers except 0 and 1
- c)All integers except 1
- d)None of these
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
Let g be the greatest integer function. Then the function f(x) = (g(x)...
g(x) = [x]
f(x) = [x]2 − [x]
f(x) is discontinuous at every integers except x = 1.
f(x) = [x]2 − [x]
f(x) is discontinuous at every integers except x = 1.
Most Upvoted Answer
Let g be the greatest integer function. Then the function f(x) = (g(x)...
Explanation:
Greatest Integer Function:
- The greatest integer function, denoted by g(x), gives the greatest integer less than or equal to x. For example, g(3.7) = 3 and g(5) = 5.
Given Function:
- The function f(x) = (g(x))^2 - g(x^2) involves the greatest integer function.
Discontinuity:
- The function f(x) will be discontinuous at points where the greatest integer function changes abruptly.
Analysis:
- Let's consider the points where g(x) and g(x^2) may change abruptly.
- At integer values of x, g(x) changes abruptly. For example, g(2.5) = 2 and g(3.5) = 3.
- At integer values of x, g(x^2) also changes abruptly. For example, g(2^2) = g(4) = 4 and g(3^2) = g(9) = 9.
Conclusion:
- The function f(x) = (g(x))^2 - g(x^2) will be discontinuous at all integer values of x except 1.
- Therefore, the correct answer is option C - "All integers except 1".
Greatest Integer Function:
- The greatest integer function, denoted by g(x), gives the greatest integer less than or equal to x. For example, g(3.7) = 3 and g(5) = 5.
Given Function:
- The function f(x) = (g(x))^2 - g(x^2) involves the greatest integer function.
Discontinuity:
- The function f(x) will be discontinuous at points where the greatest integer function changes abruptly.
Analysis:
- Let's consider the points where g(x) and g(x^2) may change abruptly.
- At integer values of x, g(x) changes abruptly. For example, g(2.5) = 2 and g(3.5) = 3.
- At integer values of x, g(x^2) also changes abruptly. For example, g(2^2) = g(4) = 4 and g(3^2) = g(9) = 9.
Conclusion:
- The function f(x) = (g(x))^2 - g(x^2) will be discontinuous at all integer values of x except 1.
- Therefore, the correct answer is option C - "All integers except 1".
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Let g be the greatest integer function. Then the function f(x) = (g(x))2− g(x2) is discontinuous ata)All integersb)All integers except 0 and 1c)All integers except 1d)None of theseCorrect answer is option 'C'. Can you explain this answer?
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