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A function f(x) is defined by f(x) =(x - 2) + x2 overall real values of x, now f(x) is
- a)continuous at x = 2
- b)discontinuous at x = 2
- c)Undefined at x = 2
- d)none of these
Correct answer is option 'A'. Can you explain this answer?
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A function f(x) is defined by f(x) =(x - 2) + x2 overall real values o...
Understanding the Function f(x)
The function f(x) is defined as:
f(x) = (x - 2) + x^2
This function is a polynomial, which is made up of basic algebraic components. Polynomials are defined for all real values of x, meaning they have no restrictions on their domain.
Continuity at x = 2
To determine if f(x) is continuous at x = 2, we need to check the following conditions:
- The function f(2) must be defined.
- The limit of f(x) as x approaches 2 must exist.
- The limit must equal the function's value at that point.
Step-by-Step Verification
- Evaluate f(2):
f(2) = (2 - 2) + 2^2 = 0 + 4 = 4
- Calculate the Limit as x Approaches 2:
lim(x → 2) f(x) = lim(x → 2) [(x - 2) + x^2]
= (2 - 2) + 2^2 = 0 + 4 = 4
- Check if the Limit Equals f(2):
lim(x → 2) f(x) = 4 = f(2)
Since all conditions are satisfied, we conclude that f(x) is continuous at x = 2.
Conclusion
- The correct answer is option 'A': f(x) is continuous at x = 2.
- This is due to the function being defined, and the limit existing and being equal to the function's value at that point.
In summary, f(x) behaves well at x = 2, confirming its continuity.
The function f(x) is defined as:
f(x) = (x - 2) + x^2
This function is a polynomial, which is made up of basic algebraic components. Polynomials are defined for all real values of x, meaning they have no restrictions on their domain.
Continuity at x = 2
To determine if f(x) is continuous at x = 2, we need to check the following conditions:
- The function f(2) must be defined.
- The limit of f(x) as x approaches 2 must exist.
- The limit must equal the function's value at that point.
Step-by-Step Verification
- Evaluate f(2):
f(2) = (2 - 2) + 2^2 = 0 + 4 = 4
- Calculate the Limit as x Approaches 2:
lim(x → 2) f(x) = lim(x → 2) [(x - 2) + x^2]
= (2 - 2) + 2^2 = 0 + 4 = 4
- Check if the Limit Equals f(2):
lim(x → 2) f(x) = 4 = f(2)
Since all conditions are satisfied, we conclude that f(x) is continuous at x = 2.
Conclusion
- The correct answer is option 'A': f(x) is continuous at x = 2.
- This is due to the function being defined, and the limit existing and being equal to the function's value at that point.
In summary, f(x) behaves well at x = 2, confirming its continuity.
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A function f(x) is defined by f(x) =(x - 2) + x2 overall real values o...
Discontinues 2
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A function f(x) is defined by f(x) =(x - 2) + x2 overall real values of x, now f(x) isa)continuous at x = 2b)discontinuous at x = 2c)Undefined at x = 2d)none of theseCorrect answer is option 'A'. Can you explain this answer?
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