Math Problem Statement
Solution
The image presents a question about the domain and range of the function where and .
The correct domain for is . This is because the logarithm function, , is defined only for positive values of . The values of must be greater than 0 to satisfy the conditions for a valid logarithmic input.
The choice in the image labeled as option B, "D = (0, \infty), because the range of its inverse function is ," is the correct answer. The reason provided is also accurate: the domain of a function is the range of its inverse, and vice versa. The inverse function of is , which indeed has a range of since yields positive outputs for any real .
Do you need any further clarification on this concept, or is there another question on this topic you would like to explore? Here are some related questions you might consider:
- What is the range of ?
- How does changing the base of the logarithm affect the graph of ?
- What are the properties of the logarithm function that are important for solving equations?
- How do you solve equations involving logarithms?
- Can you explain how to convert between different logarithmic bases?
Tip: Remember, understanding the relationship between a function and its inverse can greatly simplify solving problems related to domain and range.
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Math Problem Analysis
Mathematical Concepts
Logarithmic Functions
Inverse Functions
Domain and Range
Formulas
f(x) = log_b x
Inverse: g(x) = b^x
Theorems
Inverse Function Theorem
Suitable Grade Level
Grades 10-12