Math Problem Statement
Solution
We are given two functions and their domains:
- with the natural domain .
- We are tasked with finding the natural domain of .
Solution:
For the function , we are told that the domain of is . This implies that the argument of the function, , lies within this interval: Solving for , we divide the entire inequality by 2: Thus, .
Now, for the function , the argument must also lie within the domain . So, we have: To solve this inequality, we rewrite it as an exponential: which simplifies to: Thus, the natural domain of is .
Final Answer:
The natural domain of is .
Would you like any additional details or have any questions?
Here are 5 related questions for further exploration:
- How does changing the base of the logarithm affect the domain of a function like ?
- What is the domain of if the domain of is ?
- How does the transformation of functions affect the domain compared to ?
- What happens to the domain of compared to ?
- How does shifting the function by a constant affect its domain?
Tip: When working with logarithmic functions, remember that the argument of the logarithm must always be positive.
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Math Problem Analysis
Mathematical Concepts
Functions
Logarithms
Domain of Functions
Inequalities
Formulas
Domain of logarithmic functions
Logarithmic inequality
Base-2 logarithms
Theorems
Logarithmic function domain theorem
Basic properties of logarithms
Suitable Grade Level
Grades 11-12 or early college level