Math Problem Statement
Solution
Let's work through the problem step by step.
Domain of the function :
The natural logarithm function is only defined when its argument is positive. This means we need the expression inside the logarithm, , to be greater than zero. In other words, the domain is determined by solving the inequality:
This simplifies to:
or equivalently:
Taking the square root of both sides:
So the domain of the function is .
Checking the Statements:
-
Statement 1:
- True: Since is defined and continuous for , the function is continuous where , which is .
-
Statement 2:
- False: The domain of is limited to . It is not defined for all real numbers.
-
Statement 3:
- True: is continuous on since within this interval, , and the logarithm function is continuous for positive inputs.
-
Statement 4:
- False: This statement is misleading. The function is actually defined where (not the other way around), which is .
-
Statement 5:
- True: The expression is a polynomial, and all polynomials are continuous for all real numbers. However, the logarithm limits the domain of .
Conclusion:
- The domain of the function is .
- The correct statements are: 1, 3, and 5.
Would you like any additional details or clarification?
5 Relative Questions:
- How do you determine the domain of a logarithmic function in general?
- What are the key properties of logarithmic functions regarding continuity?
- How would the domain change if the argument of the logarithm was instead of ?
- Can the function have any vertical asymptotes? If so, where?
- How does the function behave near the endpoints of the interval ?
Tip:
Always remember to check the argument of the logarithm function. It must be positive for the function to be defined!
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Math Problem Analysis
Mathematical Concepts
Logarithmic Functions
Inequalities
Continuity
Formulas
ln(x) is defined for x > 0
81 - x^2 > 0
Theorems
Continuity of logarithmic functions
Continuity of polynomials
Suitable Grade Level
Grades 10-12