Math Problem Statement
Solution
To determine which function is not continuous on the interval , let's examine each function for continuity.
Option Analysis
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Option A:
- This is a polynomial function, and polynomial functions are continuous on the entire real line. Therefore, is continuous on .
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Option B:
- This function is a rational function, which means it will be discontinuous at values of that make the denominator zero.
- To check for points of discontinuity, we need to find the roots of the denominator . Solving this for would yield specific values where the function is undefined, leading to discontinuity at those points.
- Since there are values of that make the denominator zero, is not continuous over the entire real line.
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Option C:
- This is a composition of a trigonometric function with a linear function, both of which are continuous everywhere on the real line.
- Therefore, is continuous on .
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Option D:
- This is an exponential function in the denominator, and for any real . Thus, is defined and continuous for all real values of .
- Hence, is continuous on .
Conclusion
The function (Option B) is not continuous on because the denominator can be zero for certain values of , causing discontinuities.
Would you like further details on solving for the roots of the denominator in Option B, or do you have any other questions?
Related Questions
- How do you find points of discontinuity in rational functions?
- What methods can you use to solve a polynomial equation like ?
- Why are trigonometric functions generally continuous on the real line?
- Can exponential functions in the denominator ever lead to discontinuities?
- How does the degree of a polynomial affect its continuity?
Tip
For any rational function , check for continuity by identifying values of that make , as these points lead to discontinuities in the function.
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Math Problem Analysis
Mathematical Concepts
Continuity of Functions
Rational Functions
Polynomials
Trigonometric Functions
Exponential Functions
Formulas
Continuity of Rational Functions: \( \frac{p(x)}{q(x)} \) is discontinuous where \( q(x) = 0 \)
Theorems
Continuity Theorem for Polynomials
Continuity Theorem for Trigonometric Functions
Continuity Theorem for Exponential Functions
Suitable Grade Level
Grades 11-12
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